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A statistical field approach to capital accumulation

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Abstract

This paper presents a model of capital accumulation for a large number of heterogeneous producer–consumer agents in an exchange space in which interactions depend on agents’ positions. Agents in the exchange space are subject to both attractive and repulsive forces: exchanges drive agents closer, but crowd out more distant agents. The formalism used in this paper was developed earlier by the authors and is based on statistical field theory. It allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and microeconomic features of the individual level. Our results show that the dynamics of capital accumulation and the agents’ positions in the exchange space are correlated. Interactions in the exchange space induce phases within the system that depend on the relative strength of the repulsive force. When the repulsive force is strong, the system is in a phase of regulated exchanges. An initial central position both favours and fastens capital accumulation in average, and high levels of initial capital drive agents towards the centre. Yet, this phase displays mild competition and a broad-based although slow improvement in exchange terms. In this phase, random shocks can redistribute capital and initiate a virtuous circle of capital accumulation. When the repulsive force is low, a phase of deregulated exchanges emerges, in which capital distribution is less homogeneous and competition among agents harshens. Increased mobility accelerates capital accumulation for high initial capital producers, whereas low initial capital producers are now evicted from the exchange space as their prices and revenues deteriorate. Thus, a threshold effect appears. Above a certain level of initial capital, agents benefit from and remain in a central position. Below this level, they remain at the periphery of the exchange space.

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Notes

  1. 1.

    Given that \(X_{i}\in \left[ -1,1\right] \), the normalisation factor should depend on \(X_{i}\). A computation shows that, in the approximation of agents uniformly distributed in space, it is equal to \(2d\left( 1-\frac{\cosh \frac{ X_{i}}{d}}{\exp \left( \frac{1}{d}\right) }\right) \). This function varies slowly over \(\left[ -1,1\right] \) and can be replaced by its average, \( 2d\left( 1-\frac{\cosh \frac{X_{i}}{d}}{\exp \left( \frac{1}{d}\right) } \right) \simeq d\) for d close to 1, without impairing the results’ interpretations as shown at the end of appendix 3.

  2. 2.

    This factor could be reintroduced without impairing the results.

  3. 3.

    Due to the infinite number of possible paths, each individual path has a null probability to exist. We therefore use the word probability density rather than probability.

  4. 4.

    Actually, this paper rather considers the classical effective action, an approximation which is sufficient for the computations at stake.

  5. 5.

    The differential operator \(O\left( \Psi _{0}\left( Z,\theta \right) \right) \) can be seen as an infinite-dimensional matrix indexed by the double (infinite) entries \(\left( \underline{\left( Z,\theta \right) }^{\left[ 1 \right] },\overline{\left( Z,\theta \right) }^{\left[ 1\right] }\right) \). With this description, the kernel \(O^{-1}\left( \Psi _{0}\left( Z,\theta \right) \right) \left( \underline{\left( Z,\theta \right) }^{\left[ 1\right] },\overline{\left( Z,\theta \right) }^{\left[ 1\right] }\right) \) is the \( \left( \underline{\left( Z,\theta \right) }^{\left[ 1\right] },\overline{ \left( Z,\theta \right) }^{\left[ 1\right] }\right) \) element of the inverse matrix.

  6. 6.

    This is a necessary although not sufficient condition.

  7. 7.

    Note that the functions \(\Psi _{0}^{\left( 1\right) }\) and \(\Psi _{0}^{\left( 2\right) }\) each depend on the two variables X and K, but the dependency of \(\Psi _{0}^{\left( 1\right) }\) in K and \(\Psi _{0}^{\left( 2\right) }\) in X is of second order in \(\chi _{1}\) and \(\chi _{2}\), which justifies our notations.

  8. 8.

    The same result can be obtained by considering the contribution of \(\int \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\) in any graph. It corresponds to introducing two 2-points vertices. By convolution with Gaussian propagators, they act as the identity operator in first approximation. Hence \(\left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\) can be replaced by \(\frac{1}{2}\) to produce in average a factor of 1 after integration.

  9. 9.

    the parameter \(-\alpha _{X}\) is the lowest eigenvalue associated to the evolution operator whose G is the Green function.

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees and the editorial board for their thorough and constructive reviews, as well as for their positive comments. All errors remain ours.

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Appendices

Appendices

Appendix 1. translation from probabilistic description to field theory

This appendix summarises the most useful steps of the method developed in Gosselin et al. (2017, 2020), to switch from the probabilistic description of the model to the field-theoretic formalism. By convention and unless otherwise mentioned, the symbol \(\int \) refers to all the variables involved.

Principle

For a large number of agents, the system described by (16) involves a large number of variables \(K_{i}\left( t\right) \), \(P_{i}\left( t\right) \) and \(X_{i}\left( t\right) \) that are difficult to handle. To overcome this difficulty, we consider the space H of complex functions defined on the space of a single agent’s actions. The space H describes the collective behaviour of the system. Each function \(\Psi \) of H encodes a particular state of the system. We then associate to each function \(\Psi \) of H a statistical weight, i.e. a probability describing the state encoded in \(\Psi \). This probability is written \(\exp \left( -S\left( \Psi \right) \right) \), where \(S\left( \Psi \right) \) is a functional, i.e. the function of the function \(\Psi \). The form of \(S\left( \Psi \right) \) is derived directly from the form of (16).

The present paper’s statistical weight is a variation of the set up presented in Gosselin et al. (2020), in which a general weight describing interactions between individual agents was written, accounting for the present paper notations:

$$\begin{aligned}&\sum _{i}\int _{0}^{T}\left( \frac{1}{2\sigma ^{2}}\int _{0}^{T}\left( \frac{\hbox {d} }{\hbox {d}s}Z_{i}\left( s\right) \right) ^{2}+V_{1}\left( Z_{s}^{\left( i\right) }\right) \right) \hbox {d}s\nonumber \\&\quad +\frac{1}{2\eta ^{2}}\sum _{i}\int _{0}^{T}\left( \frac{d}{ \hbox {d}t}Z_{i}\left( t\right) -H\left( Z_{i}\left( t\right) \right) \right) ^{2}\hbox {d}t \nonumber \\&\quad +\sum _{k\geqslant 2}\sum _{i_{1},...,i_{k}}\int _{0}^{T}\int _{0}^{T}\frac{ V_{k}\left( Z_{s_{1}}^{\left( i_{1}\right) },...,Z_{s_{k}}^{\left( i_{k}\right) }\right) }{\xi ^{2}}\hbox {d}s_{1}...\hbox {d}s_{k} \end{aligned}$$
(63)

where \(Z_{i}\left( s\right) \) describes the position of agent i in a space of arbitrary economic variables. In the present paper \(Z_{i}=\left( K_{i},P_{i},X_{i}\right) \).

We showed that the field action functional:

$$\begin{aligned} S\left( \Psi \right)= & {} \int \!\!\left( \Psi ^{\dag }\left( Z\right) \!\!\left( -\frac{\sigma ^{2}}{2}\nabla ^{2}+V_{1}\left( Z\right) +\alpha \right) \Psi \left( Z\right) \right) \hbox {d}Z \nonumber \\&-\frac{1}{2}\sum _{i}\int \Psi ^{\dag }\left( Z\right) \!\!\left( \eta ^{2}\nabla ^{2}+\nabla .H\left( Z\right) \right) \Psi \left( Z\right) \hbox {d}Z\nonumber \\&+ \frac{1}{2}\sum _{i}\int \Psi ^{\dag }\left( Z\right) \left( \!\!\nabla .H\left( Z\right) \right) \Psi \left( Z\right) \hbox {d}Z \nonumber \\&+\frac{1}{\xi ^{2}}\sum _{k\geqslant 2}\sum _{k\geqslant 2}\int \Psi \left( Z_{1}\right) ...\Psi \left( Z_{k}\right) V_{k}\left( Z_{1}...Z_{k}\right) \Psi ^{\dag }\left( Z_{1}\right) ...\Psi ^{\dag }\left( Z_{k}\right) \hbox {d}Z_{1}...\hbox {d}Z_{k} \nonumber \\ \end{aligned}$$
(64)

contains the same information about the system, where \(\alpha \) is the parameter arising in the Laplace transform of the statistical weight (63), and \(\Psi ^{\dag }\left( Z\right) \) denotes the complex conjugate of \(\Psi \left( Z\right) \). The operator \(\nabla \) is the gradient operator, a vector whose i-th coordinate is the first derivative \(\frac{\partial }{ \partial Z_{i}}\): \(\nabla =\left( \frac{\partial }{\partial Z_{i}}\right) \). The operator \(\nabla ^{2}\) denotes the Laplacian:

$$\begin{aligned} \nabla ^{2}=\sum _{i}\frac{\partial ^{2}}{\partial Z_{i}^{2}} \end{aligned}$$

where the sum runs over the coordinates \(Z_{i}\) of the vector Z.

However, the translation defined by (63) and (64) does not straightforwardly apply to the present paper and has to be adapted. Actually, the interactions between different agents in the global statistical weight (16)–the terms involving sums over different agents jk, ...–are local in the time variable, i.e. the quantities involved in these terms are considered simultaneously. On the contrary, interactions are not local in (63).

The introduction of local interactions can be done by introducing a counting variable \(\Theta _{i}\left( s\right) \) for each agent i. This variable is roughly equal to s, up to some random fluctuation. This amounts to introducing a dynamic variable \(\theta \) in the field formalism to account for \(\Theta _{i}\). Thus, the field \(\Psi \left( Z\right) \) is replaced by a function \(\Psi \left( Z,\theta \right) \).

A statistical weight has to be introduced for the counting variable \(\Theta _{i}\left( s\right) \):

$$\begin{aligned} \exp \left( -\int \frac{\left( \dot{\Theta }_{i}\left( s\right) -1\right) ^{2} }{2\vartheta ^{2}}\right) \end{aligned}$$

where \(\vartheta ^{2}<<1\). This ensures that \(\Theta _{i}\left( s\right) \simeq s\), up to an initial constant. This constant can be discarded if we consider \(\Theta _{i}\left( 0\right) =0\) in all transition functions. The field counterpart of this particular weight is:

$$\begin{aligned}&\int \exp \left( -\int \frac{\left( \dot{\Theta }_{i}\left( s\right) -1\right) ^{2}}{2\vartheta ^{2}}\right) \mathcal {D}\Theta _{i}\left( s\right) \\&\quad \rightarrow \exp \left( -\Psi ^{\dag }\left( Z,\theta \right) \nabla _{\theta }.\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) \Psi \left( Z,\theta \right) \right) \end{aligned}$$

Local interactions can then be included. Assume a two-agent interaction of the type:

$$\begin{aligned} \sum _{i,j}\int V\left( Z_{i}\left( t\right) ,Z_{j}\left( t\right) \right) \hbox {d}t \end{aligned}$$
(65)

where V is an arbitrary function. We replace the time t in \(V\left( Z_{i}\left( t\right) ,Z_{j}\left( t\right) \right) \) by two independent parameters \(t_{i}\) and \(t_{j}\). We impose the equality of the counting variables associated to these parameters: \(\theta _{j}\left( t_{j}\right) -\theta _{i}\left( t_{i}\right) =0\). Consequently, the potential rewrites:

$$\begin{aligned} \sum _{i,j}\int V\left( Z_{i}\left( t_{i}\right) ,Z_{j}\left( t_{j}\right) \right) \delta \left( \theta _{j}\left( t_{j}\right) -\theta _{i}\left( t_{i}\right) \right) dt_{i}dt_{j} \end{aligned}$$
(66)

In this form, the translation from (63) to (64) can be used and yields a field-theoretic potential:

$$\begin{aligned} \int \Psi \left( Z_{1},\theta \right) \Psi \left( Z_{2},\theta \right) V\left( Z_{1},Z_{2}\right) \Psi ^{\dag }\left( Z_{1},\theta \right) \Psi ^{\dag }\left( Z_{2},\theta \right) \hbox {d}Z_{1}\hbox {d}Z_{2}d\theta \end{aligned}$$

This can be generalised straightforwardly for k interacting agents’ potentials:

$$\begin{aligned} \sum _{k\geqslant 2}\sum _{i_{1},...,i_{k}}\int _{0}^{T}\int _{0}^{T}\frac{ V_{k}\left( Z_{s_{1}}^{\left( i_{1}\right) },...,Z_{s_{k}}^{\left( i_{k}\right) }\right) }{\xi ^{2}}\hbox {d}s_{1}...\hbox {d}s_{k} \end{aligned}$$
(67)

that translate into the field-theoretic version:

$$\begin{aligned} \frac{1}{\xi ^{2}}\sum _{k\geqslant 2}\sum _{k\geqslant 2}\int \Psi \left( Z_{1},\theta \right) ...\Psi \left( Z_{k},\theta \right) V_{k}\left( Z_{1}...Z_{k}\right) \Psi ^{\dag }\left( Z_{1},\theta \right) ...\Psi ^{\dag }\left( Z_{k},\theta \right) \hbox {d}Z_{1}...\hbox {d}Z_{k}d\theta \end{aligned}$$
(68)

Translation of the model in terms of field theory

We can apply the results of the previous paragraph to the model, to translate (16) into its field theory counterpart. The statistical weights of the system can be divided into two parts, a first one for the position X in the exchange space, and a second one for K and P. We will deal separately with these two parts. In the sequel we set \(Z=\left( K,P,X\right) \) and \(Z_{i}=\left( K_{i},P_{i},X_{i}\right) \) for any index i.

Translation of (14) The X-part (14) of the statistical weight directly fits into the formalism defined by (63) and (64), (65), (66), (67) and (68). As a consequence the—log—weight (14):

$$\begin{aligned} -\sum _{i}\int \left( \frac{\left( \dot{X}_{i}\left( t\right) \right) ^{2}}{ 2\sigma _{X}^{2}}+V_{0}\left( X_{i}\left( t\right) \right) +\sum _{j}V_{1}\left( d_{ij}\left( t\right) \right) +\sum _{j,k}V_{2}\left( d_{ij}\left( t\right) ,d_{ik}\left( t\right) ,d_{jk}\left( t\right) \right) \right) \hbox {d}t \end{aligned}$$
(69)

has the following equivalent in terms of field:

$$\begin{aligned} S_{2}\left( \Psi \right)= & {} \int \Psi ^{\dag }\left( Z,\theta \right) \left( -\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}\right) \Psi \left( Z,\theta \right) +\int V_{0}\left( X\right) \left| \Psi \left( Z,\theta \right) \right| ^{2}\nonumber \\&+\int V_{1}\left( d_{12}\right) \left| \Psi \left( Z_{1},\theta \right) \right| ^{2}\left| \Psi \left( Z_{2},\theta \right) \right| ^{2} \nonumber \\&+\int V_{2}\left( d_{12},d_{13},d_{23}\right) \left| \Psi \left( Z_{1},\theta \right) \right| ^{2}\left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2} \end{aligned}$$
(70)

as reported in the text.

Translation of (12) and (13) The statistical weight for the price plus the capital part is the product of (12) with (13):

$$\begin{aligned} \exp \left( -\int \left( \frac{1}{2\sigma ^{2}}\left( \dot{K}_{i}\left( t\right) +\delta K_{i}\left( t\right) -AP_{i}\left( t\right) K_{i}^{\alpha }\left( t\right) +U_{1i}\right) ^{2}+\frac{\bar{A}^{2}}{2\sigma ^{2}}\left( P_{i}^{1+\gamma }\left( t\right) K_{i}^{\alpha }\left( t\right) -U_{2,i}\right) ^{2}\right) \hbox {d}t\right) \nonumber \\ \end{aligned}$$
(71)

with:

$$\begin{aligned} U_{1i}= & {} \frac{\kappa }{d^{2}}AP_{i}\left( t\right) K_{i}^{\alpha }\left( t\right) \sum _{j,k}\frac{P_{k}\left( t\right) }{P_{j}^{\gamma }\left( t\right) }\exp \left( -\frac{\left| X_{i}\left( t\right) -X_{j}\left( t\right) \right| }{d}-\frac{\left| X_{i}\left( t\right) -X_{k}\left( t\right) \right| }{d}\right) \\ U_{2i}= & {} \frac{\kappa }{d^{2}}\sum _{j,k}P_{j}\left( t\right) K_{j}^{\alpha }\left( t\right) P_{k}\left( t\right) \exp \left( -\frac{\left| X_{i}\left( t\right) -X_{j}\left( t\right) \right| }{d}-\frac{\left| X_{k}\left( t\right) -X_{j}\left( t\right) \right| }{d}\right) \end{aligned}$$

We will now show that the associated field action writes:

$$\begin{aligned}&\int \Psi ^{\dag }\left( Z,\theta \right) \left( -\frac{\sigma ^{2}}{2} \nabla _{K}^{2}+\frac{\left( \delta K-APK^{\alpha }\left( 1-U_{1}\right) \right) ^{2}}{2\sigma ^{2}}+\left( P^{1+\gamma }K^{\alpha }+U_{2}\right) ^{2}\right. \nonumber \\&\left. \quad -\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}+\frac{1}{2\vartheta ^{2}}+\alpha \right) \Psi \left( Z,\theta \right) \end{aligned}$$
(72)

with:

$$\begin{aligned} U_{1}\left( Z,\theta \right)= & {} \frac{\kappa }{d^{2}}\int \frac{P_{3}\exp \left( -\left( \frac{d_{12}+d_{13}}{d}\right) \right) }{P_{2}^{\gamma }} \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \quad \end{aligned}$$
(73)
$$\begin{aligned} U_{2}\left( Z,\theta \right)= & {} -\frac{\kappa }{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\left( \frac{d_{12}+d_{23}}{d} \right) \right) \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\nonumber \\&\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}d\theta \end{aligned}$$
(74)

where we set \(X_{1}=X\) in \(d_{12}\) and \(d_{13}\).

To prove (72), we decompose (71) into its two components.

Translation of the second part of (71) Using (65) and (66), the second part of (71):

$$\begin{aligned} \exp \left( -\int \frac{\bar{A}^{2}}{2\sigma ^{2}}\left( P_{i}^{1+\gamma }\left( t\right) K_{i}^{\alpha }\left( t\right) -U_{2,i}\right) ^{2}\hbox {d}t\right) \end{aligned}$$

has a direct equivalent in terms of field theory:

$$\begin{aligned} \int \Psi ^{\dag }\left( K,X,\theta \right) \left( \left( P^{1+\gamma }K^{\alpha }+U_{2}\right) ^{2}\right) \Psi \left( K,X,\theta \right) \end{aligned}$$
(75)

Translation of the first part of (71) The first part of (71) is given by:

$$\begin{aligned} \exp \left( -\frac{1}{2\sigma ^{2}}\int \left( \left( \dot{K}_{i}\left( t\right) +\delta K_{i}\left( t\right) -AP_{i}\left( t\right) K_{i}^{\alpha }\left( t\right) +U_{1i}\right) ^{2}\right) \hbox {d}t\right) \end{aligned}$$
(76)

To find the field equivalent of (76), we rewrite the above expression in a more convenient form.

First, note that (76) can be written in a general form:

$$\begin{aligned} \exp \left( -\int \frac{\left( A\dot{Z}_{i}\left( t\right) +G\left( Z_{i}\left( t\right) \right) -\sum _{\left( j,n\right) }V\left( Z_{i}\left( t\right) ,\left( Z\left( t\right) \right) _{\left( j,n\right) }\right) \right) ^{2}}{2\sigma ^{2}}\right) \end{aligned}$$
(77)

where \(Z_{i}\) is a vector variable of arbitrary dimension l, \(\left( j,n\right) \) is a sequence of n indices \(\left( j_{1},j_{2},...,j_{n}\right) \) and \(\left( Z\left( t\right) \right) _{\left( j,n\right) }=\left( Z_{j_{1}}\left( t\right) ,Z_{j_{2}}\left( t\right) ,...Z_{j_{n}}\left( t\right) \right) \) where n is given (in this paper \( n=2 \) for \(U_{1}\)). The matrix A is of dimension \(1\times l\). Actually, we recover (76) when we set:

$$\begin{aligned} Z_{i}\left( t\right)= & {} \left( K_{i}\left( t\right) ,P_{i}\left( t\right) ,X_{i}\left( t\right) \right) ^{t} \nonumber \\ G\left( Z_{i}\left( t\right) \right)= & {} \delta K_{i}\left( t\right) -AP_{i}\left( t\right) K_{i}^{\alpha }\left( t\right) \nonumber \\ V\left( Z_{i}\left( t\right) ,Z_{j}\left( t\right) ,Z_{j}\left( t\right) \right)= & {} -\frac{\kappa }{d^{2}}AP_{i}\left( t\right) K_{i}^{\alpha }\left( t\right) \frac{P_{k}\left( t\right) }{P_{j}^{\gamma }\left( t\right) }\nonumber \\&\exp \left( -\frac{\left| X_{i}\left( t\right) -X_{j}\left( t\right) \right| }{d}-\frac{\left| X_{i}\left( t\right) -X_{k}\left( t\right) \right| }{d}\right) \nonumber \\ A= & {} \left( \begin{array}{ccc} 1&0&0 \end{array} \right) \end{aligned}$$
(78)

In the sequel, \(\dot{Z}_{i}\left( t\right) \) will stand for \(A\dot{Z} _{i}\left( t\right) \). This is done for the sake of simplicity and does not impair the argument.

Second, as explained in the translation from (65) to (66), we can replace the time t in \(V\left( Z_{i}\left( t\right) ,\left( Z\left( t\right) \right) _{\left( j,n\right) }\right) \) by the independent parameters \(t_{i}\) and \(t_{\left( n\right) }=\left( t_{1},...,t_{n}\right) \) . We write:

$$\begin{aligned} \left( Z\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }=\left( Z_{j_{1}}\left( t_{1}\right) ,Z_{j_{2}}\left( t_{2}\right) ,...Z_{j_{n}}\left( t_{n}\right) \right) \end{aligned}$$

We then impose the equality of the counting variables associated to these parameters: \(\theta _{j_{l}}\left( t_{l}\right) -\theta _{i}\left( t_{i}\right) =0\) for \(l=1,...,n\). Consequently, the weight (77) rewrites:

$$\begin{aligned} \exp \left( -\int \frac{\left( \dot{Z}_{i}\left( t_{i}\right) +G\left( Z_{i}\left( t_{i}\right) \right) -\sum _{\left( j,n\right) }\int V\left( Z_{i}\left( t_{i}\right) ,\left( Z\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \delta \left( \left( \theta \left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }-\theta _{i}\left( t_{i}\right) \right) \right) ^{2}\prod \limits _{l=1}^{n}dt_{l}}{2\sigma ^{2}} dt_{i}\right) \nonumber \\ \end{aligned}$$
(79)

with:

$$\begin{aligned} \delta \left( \left( \theta \left( t_{\left( n\right) }\right) _{\left( j,n\right) }\right) -\theta _{i}\left( t_{i}\right) \right) =\prod \limits _{l=1}^{n}\delta \left( \left( \theta _{j_{l}}\left( t_{l}\right) \right) -\theta _{i}\left( t_{i}\right) \right) \end{aligned}$$

To simplify notations and account for possible generalisations, we replace the potential in (79):

$$\begin{aligned} V\left( Z_{i}\left( t_{i}\right) ,\left( Z\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \delta \left( \left( \theta \left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }-\theta _{i}\left( t_{i}\right) \right) \end{aligned}$$

by a more general form, also denoted V for the sake of simplicity:

$$\begin{aligned}&\int V\left( Z_{i}\left( t_{i}\right) ,\left( Z\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \delta \left( \left( \theta \left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }-\theta _{i}\left( t_{i}\right) \right) \\&\quad \rightarrow \int V\left( Z_{i}\left( t_{i}\right) ,\theta _{i}\left( t_{i}\right) ,\left( Z\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) },\left( \theta \left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \\&\quad \equiv \int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \end{aligned}$$

where we set:

$$\begin{aligned} Y_{i}\left( t_{i}\right)= & {} \left( Z_{i}\left( t_{i}\right) ,\theta _{i}\left( t_{i}\right) \right) \\ \left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }= & {} \left( Y_{j_{1}}\left( t_{1}\right) ,\theta _{j_{1}}\left( t_{1}\right) ,Y_{j_{2}}\left( t_{2}\right) ,\theta _{j_{2}}\left( t_{2}\right) ...\right) \end{aligned}$$

so that the \(\delta \) factor in the potential is replaced by an arbitrary function of the counting variables \(\theta _{i}\left( t_{i}\right) \) and \( \left( \theta \left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\).

Third, we modify (77) by introducing an auxiliary variable \(\tilde{Z }_{i}\left( t_{i}\right) \), equal to \(\sum _{\left( j,n\right) }\int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \), up to a random error of square deviation \( \sigma _{2}^{2}<<1\). Ultimately, the weight (77) becomes:

$$\begin{aligned}&\exp \left( -\frac{\left( \dot{Z}_{i}\left( t_{i}\right) +G\left( Z_{i}\left( t_{i}\right) \right) -\sum _{\left( j,n\right) }\int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \right) ^{2}}{2\sigma ^{2}}\right) \nonumber \\&\quad \simeq \exp \left( -\frac{\left( \dot{Z}_{i}\left( t\right) +G\left( Z_{i}\left( t_{i}\right) \right) -\tilde{Z}_{i}\left( t_{i}\right) \right) ^{2}}{2\sigma ^{2}}-\frac{\left( \tilde{Z}_{i}\left( t_{i}\right) -\sum _{\left( j,n\right) }\int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \right) ^{2}}{2\sigma _{2}^{2}}\right) \nonumber \\&\quad =\exp \left( -\frac{\left( \dot{Z}_{i}\left( t\right) +G\left( Z_{i}\left( t_{i}\right) \right) -\tilde{Z}_{i}\left( t_{i}\right) \right) ^{2}}{2\sigma ^{2}}\right. \nonumber \\&\qquad \left. -\frac{\left( \tilde{Z}_{i}^{2}\left( t_{i}\right) \right) ^{2}-2 \tilde{Z}_{i}\left( t_{i}\right) \sum _{\left( j,n\right) }\int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) +\left( \sum _{\left( j,n\right) }\int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \right) ^{2}}{2\sigma _{2}^{2}}\right) \nonumber \\ \end{aligned}$$
(80)

Actually, the condition \(\sigma _{2}^{2}<<1\) ensures that states with \( \tilde{Z}_{i}\left( t_{i}\right) -\sum _{\left( j,n\right) }\int V\left( Y_{i}\left( t_{i}\right) ,\left( Y\left( t_{\left( n\right) }\right) \right) _{\left( j,n\right) }\right) \ne 0\) have a negligible probability.

We also assume that:

$$\begin{aligned} \sigma _{2}^{2}<<\sigma ^{2} \end{aligned}$$

The techniques of the previous paragraph apply, since the weight (80) has the form required. The corresponding field theory action writes:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( -\nabla _{\theta }\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) -\frac{1}{2} \nabla _{Z}\left( \sigma ^{2}\nabla _{Z}+2\left( G\left( Z\right) -\tilde{Z} \right) \right) \right. \nonumber \\&\left. \quad +\frac{G^{\prime }\left( Z\right) }{2}-\frac{1}{2} \varepsilon ^{2}\nabla _{\tilde{Z}}^{2}-\frac{1}{2}\omega \tilde{Z} ^{2}\right) \Psi \left( Y,\tilde{Z}\right) \nonumber \\&\quad +\int \frac{\tilde{Z}^{2}}{2\sigma _{2}^{2}}\left| \Psi \left( Y, \tilde{Z}\right) \right| ^{2}-\int \frac{\tilde{Z}V\left( \left( Y, \tilde{Z}\right) ,\left( Y^{\prime },\tilde{Z}^{\prime }\right) _{\left( n\right) }\right) }{\sigma _{2}^{2}}\nonumber \\&\quad \left| \Psi \left( Y,\tilde{Z} \right) \right| ^{2}\left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2} \nonumber \\&\quad +\int \frac{V\left( \left( Y,\tilde{Z}\right) ,\left( Y^{\prime },\tilde{Z} ^{\prime }\right) _{\left( n\right) }\right) V\left( \left( Y,\tilde{Z} \right) ,\left( Y^{''},\tilde{Z}''\right) _{\left( n\right) }\right) }{2\sigma _{2}^{2}}\left| \Psi \left( Y,\tilde{Z}\right) \right| ^{2}\left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2} \left| \Psi \left( Y^{''},\tilde{Z} ''\right) \right| _{\left( n\right) }^{2} \nonumber \\ \end{aligned}$$
(81)

where \(\left( Y^{\prime },\tilde{Z}^{\prime }\right) _{\left( n\right) }\) is a multiplet of variables \(\left( Y_{1}^{\prime },\tilde{Z}_{1}^{\prime },Y_{2}^{\prime },\tilde{Z}_{2}^{\prime },...\right) \) and:

$$\begin{aligned} \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\equiv \prod \limits _{l=1}^{n}\left| \Psi \left( Y_{l}^{\prime },\tilde{Z}_{l}^{\prime }\right) \right| ^{2} \end{aligned}$$

This action can be further simplified by applying the following transformation on the field:

$$\begin{aligned} \Psi \left( Y,\tilde{Z}\right)= & {} \exp \Bigg ( -\int \frac{\left( G\left( Z\right) -\tilde{Z}\right) }{\sigma ^{2}}\Bigg ) \bar{\Psi }\left( Y,\tilde{Z} \right) \nonumber \\ \Psi ^{\dagger }\left( Y,\tilde{Z}\right)= & {} \exp \Bigg ( \int \frac{\left( G\left( Z\right) -\tilde{Z}\right) }{\sigma ^{2}}\Bigg ) \bar{\Psi }^{\dagger }\left( Y,\tilde{Z}\right) \end{aligned}$$
(82)

and by using a change of notation for the sake of simplicity:

$$\begin{aligned} \bar{\Psi }\left( Y,\tilde{Z}\right)\rightarrow & {} \Psi \left( Y,\tilde{Z} \right) \\ \bar{\Psi }^{\dagger }\left( Y,\tilde{Z}\right)\rightarrow & {} \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \end{aligned}$$

Equation (81) can then be replaced by the following action:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( -\nabla _{\theta }\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) -\frac{\sigma ^{2} }{2}\nabla _{Z}^{2}+\frac{\left( G\left( Z\right) -\tilde{Z}\right) ^{2}}{ 2\sigma ^{2}}-\frac{1}{2}\varepsilon ^{2}\nabla _{\tilde{Z}}^{2}-\frac{1}{2} \omega \tilde{Z}^{2}\right) \Psi \left( Y,\tilde{Z}\right) \nonumber \\&\quad +\int \frac{\tilde{Z}^{2}}{2\sigma _{2}^{2}}\left| \Psi \left( Y, \tilde{Z}\right) \right| ^{2}-\int \frac{\tilde{Z}V\left( \left( Y, \tilde{Z}\right) ,\left( Y^{\prime },\tilde{Z}^{\prime }\right) _{\left( n\right) }\right) }{\sigma _{2}^{2}} \left| \Psi \left( Y,\tilde{Z} \right) \right| ^{2}\left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2} \nonumber \\&\quad +\int \frac{V\left( \left( Y,\tilde{Z}\right) ,\left( Y^{\prime },\tilde{Z} ^{\prime }\right) _{\left( n\right) }\right) V\left( \left( Y,\tilde{Z} \right) ,\left( Y^{''},\tilde{Z}''\right) _{\left( n\right) }\right) }{2\sigma _{2}^{2}}\left| \Psi \left( Y,\tilde{Z}\right) \right| ^{2}\left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2} \left| \Psi \left( Y^{''},\tilde{Z} ''\right) \right| _{\left( n\right) }^{2}\nonumber \\ \end{aligned}$$
(83)

For any potential satisfying \(V\left( \left( Y,\tilde{Z}\right) ,\left( Y^{\prime },\tilde{Z}^{\prime }\right) _{\left( n\right) }\right) =V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \)—which is the case considered in this paper, (83) simplifies as:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( -\nabla _{\theta }\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) -\frac{\sigma ^{2} }{2}\nabla _{Z}^{2}+\frac{\left( G\left( Z\right) -\tilde{Z}\right) ^{2}}{ 2\sigma ^{2}}-\frac{1}{2}\varepsilon ^{2}\nabla _{\tilde{Z}}^{2}-\frac{1}{2} \omega \right) \Psi \left( Y,\tilde{Z}\right) \nonumber \\&\quad +\int \frac{\left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z} ^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma _{2}^{2}}\left| \Psi \left( Y,\tilde{Z}\right) \right| ^{2} \end{aligned}$$
(84)

This formula expresses (79), and as a consequence also (80), in terms of field theory.

However, it further simplifies since, given our assumptions, \(\varepsilon ^{2}<<1,\sigma _{2}^{2}<<1\). We thus consider that: \(\varepsilon ^{2}\nabla _{\tilde{Z}}^{2}\rightarrow 0\), which implies that the condition:

$$\begin{aligned} 0=\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}\Psi \left( Y,\tilde{Z}\right) \end{aligned}$$

imposed by \(\sigma _{2}^{2}<<1\) in (84) is obtained for a function of the type:

$$\begin{aligned} \Psi \left( Y,\tilde{Z}\right) =\delta \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \Psi \left( Y\right) \end{aligned}$$
(85)

The function \(\delta \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \) can be interpreted as a Gaussian function of norm equal to 1, and peaked around \(\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime }\right) \right| _{\left( n\right) }^{2}\).

Expression (84) can thus be written for fields of the form (85). In fact, Eq. (85) implies that the term arising in (84):

$$\begin{aligned} \int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \frac{\left( G\left( Z\right) -\tilde{Z}\right) ^{2}}{2\sigma ^{2}}\Psi \left( Y,\tilde{Z}\right) \end{aligned}$$
(86)

can be neglected. Actually:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) - \tilde{Z}\right) ^{2}\Psi \left( Y,\tilde{Z}\right) \nonumber \\&\quad =\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}\Psi \left( Y,\tilde{Z}\right) \nonumber \\&\qquad -2\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \nonumber \\&\qquad \times \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \Psi \left( Y,\tilde{Z} \right) \nonumber \\&\qquad +\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}\Psi \left( Y,\tilde{Z}\right) \nonumber \\&\quad =\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}\Psi \left( Y,\tilde{Z}\right) \nonumber \\&\qquad -2\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \nonumber \\&\qquad \times \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \Psi \left( Y,\tilde{Z} \right) \end{aligned}$$
(87)

Now remark that the last quantity in (87) has a negligible norm. Actually:

$$\begin{aligned}&\left| \int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \right. \nonumber \\&\left. \qquad \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \Psi \left( Y,\tilde{Z}\right) \right| \nonumber \\&\quad \leqslant \left\| \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \right\| \nonumber \\&\qquad \times \left\| \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime }, \tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \Psi \left( Y,\tilde{Z}\right) \right\| \end{aligned}$$
(88)

and the norm of the last factor in (88) is close to zero, since:

$$\begin{aligned}&\left\| \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z} ^{\prime }\right) \right| _{\left( n\right) }^{2}\right) \Psi \left( Y, \tilde{Z}\right) \right\| ^{2} \nonumber \\&\quad =\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( \tilde{Z}-\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}\Psi \left( Y,\tilde{Z}\right) \simeq 0 \nonumber \\ \end{aligned}$$
(89)

Ultimately, Eqs. (88) and (89) imply that (86) is negligible.

As a consequence, the potential (84) simplifies as:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \left( -\nabla _{\theta }\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) -\frac{\sigma ^{2} }{2}\nabla _{Z}^{2}\right) \Psi \left( Y,\tilde{Z}\right) \\&\qquad +\int \Psi ^{\dagger }\left( Y,\tilde{Z}\right) \frac{\left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}}\Psi \left( Y, \tilde{Z}\right) \\&\quad =\int \Psi ^{\dagger }\left( Y\right) \left( -\frac{\sigma ^{2}}{2}\nabla _{Z}^{2}+\frac{\left( G\left( Z\right) -\int V\left( Y,\left( Y^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Y^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}} \right) \Psi \left( Y\right) \end{aligned}$$

The above equation is the field equivalent of (80). We can now come back to the initial variables by letting \(Y=\left( Z,\theta \right) \). This ultimately yields the following field-theoretic translation of (77):

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Z,\theta \right) \left( -\nabla _{\theta }\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) -\frac{\sigma ^{2} }{2}\nabla _{Z}^{2}\right. \nonumber \\&\left. \quad +\frac{\left( G\left( Z\right) -\int V\left( Z,\left( Z^{\prime }\right) _{\left( n\right) }\right) \delta \left( \theta -\theta ^{\prime }\right) \left| \Psi \left( Z^{\prime },\theta ^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}} \right) \Psi \left( Z,\theta \right) \nonumber \\&\quad = \int \Psi ^{\dagger }\left( Z,\theta \right) \left( -\nabla _{\theta }\left( \frac{\vartheta ^{2}}{2}\nabla _{\theta }-1\right) -\frac{\sigma ^{2} }{2}\nabla _{Z}^{2}\right. \nonumber \\&\left. \qquad +\frac{\left( G\left( Z\right) -\int V\left( Z,\left( Z^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Z^{\prime },\theta ^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}}\right) \Psi \left( Z,\theta \right) \end{aligned}$$
(90)

Translation of (71) We now gather (75) and (90) to obtain the field equivalent of (71). We perform a change of variable in the counting variable:

$$\begin{aligned} \Psi \left( Z,\theta \right)= & {} \exp \left( \frac{\theta }{\vartheta ^{2}} \right) \bar{\Psi }\left( Z,\theta \right) \\ \Psi ^{\dagger }\left( Z,\theta \right)= & {} \exp \left( \frac{\theta }{ \vartheta ^{2}}\right) \bar{\Psi }^{\dagger }\left( Z,\theta \right) \end{aligned}$$

and again reset:

$$\begin{aligned} \bar{\Psi }\left( Z,\theta \right)\rightarrow & {} \Psi \left( Z,\theta \right) \\ \bar{\Psi }^{\dagger }\left( Z,\theta \right)\rightarrow & {} \Psi ^{\dagger }\left( Z,\theta \right) \end{aligned}$$

Given our choices for G and V in (78), we obtain (72).

Green functions

To conclude this section, we includethe contribution of the change of variable (82) to the computation of the Green functions. Because of the change of variable, the source terms must be included from the beginning. This leads to the following action plus source terms:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Z,\tilde{Z}\right) \left( -\frac{\vartheta ^{2} }{2}\nabla _{\theta }^{2}+\frac{1}{2\vartheta ^{2}}-\frac{\sigma ^{2}}{2} \nabla _{Z}^{2}\right) \Psi \left( Z,\tilde{Z}\right) \\&\qquad +\int \Psi ^{\dagger }\left( Z,\tilde{Z}\right) \frac{\left( G\left( Z\right) -\int V\left( Z,\left( Z^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Z^{\prime },\tilde{Z}^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}}\Psi \left( Z, \tilde{Z}\right) \\&\qquad +\int J\left( Z\right) \exp \left( -\int ^{Z}\frac{\left( G\left( Z\right) - \tilde{Z}\right) }{\sigma ^{2}}\right) \Psi \left( Z,\tilde{Z}\right) +J^{\dagger }\left( Z\right) \exp \left( \int ^{Z}\frac{\left( G\left( Z\right) -\tilde{Z}\right) }{\sigma ^{2}}\right) \Psi ^{\dagger }\left( Z, \tilde{Z}\right) \\&\quad =\int \Psi ^{\dagger }\left( Z\right) \left( -\frac{\vartheta ^{2}}{2} \nabla _{\theta }^{2}+\frac{1}{2\vartheta ^{2}}-\frac{\sigma ^{2}}{2}\nabla _{Z}^{2}+\frac{\left( G\left( Z\right) +\int V\left( Z,\left( Z^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Z^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}} \right) \Psi \left( Z\right) \\&\qquad +\int J\left( Z\right) \exp \left( -\int ^{Z}\frac{\left( G\left( Z\right) - \tilde{Z}\right) }{2}\right) \Psi \left( Z\right) +J^{\dagger }\left( Z\right) \exp \left( \int ^{Z}\frac{\left( G\left( Z\right) -\tilde{Z}\right) }{2}\right) \Psi ^{\dagger }\left( Z\right) \end{aligned}$$

Given (85), (78) and (73), the value of \(\tilde{Z}\) is:

$$\begin{aligned} \tilde{Z}\left( Z\right)= & {} \int V\left( Z,\left( Z^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Z^{\prime }\right) \right| _{\left( n\right) }^{2} \\= & {} -U_{1}\left( Z\right) \end{aligned}$$

Consequently, the action including the source terms writes:

$$\begin{aligned}&\int \Psi ^{\dagger }\left( Z\right) \left( -\frac{\vartheta ^{2}}{2} \nabla _{\theta }^{2}+\frac{1}{2\vartheta ^{2}}-\frac{\sigma ^{2}}{2}\nabla _{Z}^{2}\right) \Psi \left( Z\right) \\&\quad +\int \Psi ^{\dagger }\left( Z\right) \frac{\left( G\left( Z\right) -\int V\left( Z,\left( Z^{\prime }\right) _{\left( n\right) }\right) \left| \Psi \left( Z^{\prime }\right) \right| _{\left( n\right) }^{2}\right) ^{2}}{2\sigma ^{2}}\Psi \left( Z\right) \\&\quad +\int J\left( Z\right) \exp \left( -\int ^{Z}\frac{\left( G\left( Z\right) +U_{1}\left( Z\right) \right) }{\sigma ^{2}}\right) \Psi \left( Z\right) \\&\quad +\int J^{\dagger }\left( Z\right) \exp \left( \int ^{Z}\frac{\left( G\left( Z\right) +U_{1}\left( Z\right) \right) }{\sigma ^{2}}\right) \Psi ^{\dagger }\left( Z\right) \end{aligned}$$

The Green functions are then computed through the following formulas:

$$\begin{aligned} G\left( Z,Y\right)= & {} \left\langle \exp \left( -\left( \int ^{Z}\frac{\left( G\left( Z\right) +U_{1}\left( Z\right) \right) }{\sigma ^{2}}\right) \right) \right. \\&\times \left. \Psi \left( Z\right) \Psi ^{\dagger }\left( Y\right) \exp \left( \left( \int ^{Y}\frac{\left( G\left( Z\right) +U_{1}\left( Z\right) \right) }{\sigma ^{2}}\right) \right) \right\rangle \\= & {} \left\langle \exp \left( -\left( \int _{Y}^{Z}\frac{\left( G\left( Z\right) +U_{1}\left( Z\right) \right) }{\sigma ^{2}}\right) \right) \Psi \left( Z\right) \Psi ^{\dagger }\left( Y\right) \right\rangle \end{aligned}$$

Appendix 2. Condition for a non-trivial phase

We find the conditions under which a non-trivial phase appears for the system by inspecting the configurations \(\Psi _{0}\left( Z,\theta \right) \) minimising the action.

In the absence of any dynamics for X, and since the potential for (KP) is positive, the minimal configuration is null. The possibility of non-trivial configurations thus depends on the X-part of the action (22). Recall that we set \(Z_{i}=\left( K_{i},P_{i},X_{i}\right) \) for any index i. Using a first approximation for \(S_{2}\left( \Psi \right) \), the minimisation of (22) yields:

$$\begin{aligned} 0= & {} -\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}\Psi \left( Z,\theta \right) +V_{0}\left( X\right) \Psi \left( Z,\theta \right) +\left( \int V_{1}\left( X-X_{2}\right) \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\right) \Psi \left( Z,\theta \right) \\&+\left( \int V_{2}\left( X-X_{2},X-X_{3},d_{23}\right) \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\right) \Psi \left( Z,\theta \right) \end{aligned}$$

The condition for a minimum can be found by replacing some of these quantities by their averages:

$$\begin{aligned} \left\langle X\right\rangle= & {} 0 \\ \left| X-X_{i}\right|\simeq & {} \sqrt{\left\langle \left( X-X_{i}\right) ^{2}\right\rangle }\simeq \sqrt{2\left\langle X^{2}\right\rangle }\simeq \sqrt{2}\kappa _{0}^{-\frac{1}{4}}\sigma _{X}\\ V_{1}\left( \left| X-X_{2}\right| \right)= & {} -\frac{\kappa _{1}}{2} \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-X_{2}\right| \right) }{\left\langle K\right\rangle ^{2}}\exp \left( -\chi _{1}\left| X-X_{2}\right| \right) \\\simeq & {} -\frac{\kappa _{1}}{2}\frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-X_{2}\right| \right) }{\left\langle K\right\rangle ^{2}} \exp \left( -\chi _{1}\sqrt{2}\kappa _{0}^{-\frac{1}{4}}\sigma _{X}\right) \equiv -\frac{\bar{\kappa }_{1}}{2} \end{aligned}$$

and similarly:

$$\begin{aligned} V_{2}\left( X-X_{2},X-X_{3},d_{23}\right) \simeq \frac{\kappa _{2}}{3}\exp \left( -\chi _{2}3\sqrt{2}\kappa _{0}^{-\frac{1}{4}}\sigma _{X}\right) \equiv \frac{\bar{\kappa }_{2}}{3} \end{aligned}$$

In the sequel, the parameters \(\chi _{1}\), \(\chi _{2}\) and \(\kappa _{0}\) will be considered to be relatively small. Since we are only interested in finding an approximate condition for the existence of a non-trivial phase, we can set \(\bar{\kappa }_{1}\simeq \kappa _{1}\) and \(\bar{\kappa }_{2}\simeq \kappa _{2}\) . We are thus left with:

$$\begin{aligned}&0=-\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}\Psi \left( Z,\theta \right) + \frac{\kappa _{0}}{2\sigma _{X}^{2}}X^{2}\Psi \left( Z,\theta \right) \nonumber \\&\quad -\kappa _{1}\left( \int \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\right) \Psi \left( Z,\theta \right) +\kappa _{2}\left( \int \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\right) \Psi \left( Z,\theta \right) \nonumber \\ \end{aligned}$$
(91)

We assume a fundamental state of the form:

$$\begin{aligned} \Psi _{0}\left( Z,\theta \right) =\Psi _{0}^{\left( 1\right) }\left( X\right) \Psi _{0}^{\left( 2\right) }\left( K\right) \Psi _{0}^{\left( 3\right) }\left( P\right) \end{aligned}$$
(92)

that will be justified later on. We also define:

$$\begin{aligned} \rho ^{2}=\int \left| \Psi _{0}\left( Z,\theta \right) \right| ^{2} \end{aligned}$$
(93)

Equation (91) rewrites:

$$\begin{aligned} 0=-\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}\Psi \left( Z,\theta \right) + \frac{\kappa _{0}}{2\sigma _{X}^{2}}X^{2}\Psi \left( Z,\theta \right) -\kappa _{1}\rho ^{2}\Psi \left( Z,\theta \right) +\kappa _{2}\rho ^{4}\Psi \left( Z,\theta \right) \end{aligned}$$
(94)

This is the eigenstate equation for an harmonic oscillator in a constant potential \(-\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}\). The harmonic oscillator part has eigenvalues \(\left( \frac{1}{2}+n\right) \kappa _{0}^{ \frac{1}{2}}\) for \(n\in \mathbb {N} \). The condition for a solution of (94) with a finite norm is obtained by considering the fundamental (\(n=0\)) eigenvalue, so that (94) satisfies:

$$\begin{aligned} \frac{1}{2}\kappa _{0}^{\frac{1}{2}}-\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}=0 \end{aligned}$$
(95)

Thus, for \(\kappa _{1}^{2}-2\kappa _{0}^{\frac{1}{2}}\kappa _{2}<0\), the system has only one—trivial—phase \(\rho =0\), i.e. \(\Psi \left( Z,\theta \right) =0\). However, for \(\kappa _{1}^{2}-2\kappa _{0}^{\frac{1}{2}}\kappa _{2}>0\), there is a possibility of non-trivial phase with:

$$\begin{aligned} \rho ^{2}\simeq \frac{\kappa _{1}+\sqrt{\kappa _{1}^{2}-2\kappa _{0}^{\frac{1 }{2}}\kappa _{2}}}{2\kappa _{2}} \end{aligned}$$

The value of \(\rho \) will be refined in appendix 4.

As a consequence, the possibility of a non-trivial phase depends on the relative strength of the repulsive force over the attractive one. A non-trivial phase is possible only for a repulsive force that is strong enough.

Appendix 3. Determination of prices and average capital as functions of \(\Psi \) for \(\rho =0\)

Let us consider the phase \(\rho =0\), i.e. the case of the trivial background field \(\left| \Psi _{0}\left( Z,\theta \right) \right| \) (see (92) and (93)). We look for a configuration satisfying the market clearing condition. To do so, we replace:

$$\begin{aligned} \Psi \left( K,P,X,\theta \right) \rightarrow \delta \left( P-P\left( K,X,\theta \right) \right) \Psi \left( K,X,\theta \right) \end{aligned}$$
(96)

When needed, we will set \(Z=\left( K,P,X,\theta \right) \). We will first consider the case \(\bar{A}>>A\), before considering the corrections in \(\frac{ A}{\bar{A}}\).

Determination of prices and average capital as functions of \(\Psi \), expression of the potential. Case \(\bar{A}\rightarrow \infty \)

Defining equations as functions of the field

We determine simultaneously the price P as a function of K and X, and the average capital \(\left\langle K\right\rangle _{X}\). To do so, we consider the case where \(\bar{A}\rightarrow \infty \). We write the potential term in (20) at the zeroth order in \(\frac{A}{\bar{A}}\). Incidentally, the overall factor \(\frac{1}{2\sigma ^{2}}\) is irrelevant in this section and will be omitted.

$$\begin{aligned} \left( \delta K-APK^{\alpha }\left( 1-\frac{\kappa }{d^{2}}\int \frac{ P_{3}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) }{P_{2}^{\gamma }} \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}\right) \right) ^{2} \end{aligned}$$
(97)

with the associated constraint:

$$\begin{aligned}&P^{1+\gamma }K^{\alpha }=\frac{\kappa }{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+ \frac{\left| X_{2}-X_{3}\right| }{d}\right) \right) \nonumber \\&\quad \times \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{aligned}$$
(98)

The constraint is solved by using the field configuration (96) for a trial function P given by:

$$\begin{aligned} P=P\left( K,X\right) =\left( K\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X\right) \end{aligned}$$
(99)

Replacing this expression in the constraint (98) leads in average to the relation:

$$\begin{aligned}&\left( f\left( X\right) \right) ^{1+\gamma }=\frac{\kappa }{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{\left| X_{2}-X_{3}\right| }{d}\right) \right) \nonumber \\&\quad \times \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{aligned}$$
(100)

The potential (97) for K then becomes:

$$\begin{aligned}&\left( \delta K-A\left( K\right) ^{\frac{\alpha \gamma }{1+\gamma } }f\left( X\right) \left( 1-\frac{\kappa }{d^{2}}\int \frac{\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \exp \left( -\left( \frac{\left| X-X_{2}\right| +\left| X-X_{3}\right| }{ d}\right) \right) \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} }{\left( \left( K_{2}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{2}\right) \right) ^{\gamma }}\right) \right) ^{2} \nonumber \\&\quad =\left( \delta K-A\left( K\right) ^{\frac{\alpha \gamma }{1+\gamma } }f\left( X\right) \left( \begin{array}{c} 1-\frac{\kappa }{d^{2}}\int \left( K_{2}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma } \\ \times \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{array} \right) \right) ^{2} \end{aligned}$$
(101)

For \(\alpha \gamma<<1\), \(\left( K\right) ^{\frac{\alpha \gamma }{1+\gamma } }\simeq \left\langle K\right\rangle _{X}^{\frac{\alpha \gamma }{1+\gamma }}\) , where \(\left\langle K\right\rangle _{X}\) is defined by (31). Adding the kinetic part for K in (20) to (101) and approximating the parenthesis in (101) by its average defined in (31), we obtain for a given X, the action for an Euclidian harmonic oscillator:

$$\begin{aligned}&\int \Psi ^{\dag }\left( Z,\theta \right) \left( -\sigma ^{2}\nabla _{K}^{2}\right. \\&\quad \left. +\left( \delta K-A\left( K\right) ^{\frac{\alpha \gamma }{1+\gamma } }f\left( X\right) \times \right. \right. \\&\left. \left. \quad \times \left\langle \begin{array}{c} 1-\frac{\kappa }{d^{2}}\int \left( K_{2}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma } \\ \times \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{array} \right\rangle \right) ^{2}\right) \Psi \left( Z,\theta \right) \end{aligned}$$

whose average \(\left\langle K\right\rangle _{X}\) satisfies:

$$\begin{aligned}&\delta \left\langle K\right\rangle _{X}-A\left( \left\langle K\right\rangle _{X}\right) ^{\frac{\alpha \gamma }{1+\gamma }}f\left( X\right) \times \nonumber \\&\quad \times \left\langle \begin{array}{c} 1-\frac{\kappa }{d^{2}}\int \left( K_{2}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\times \\ \times \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{array} \right\rangle =0 \nonumber \\ \end{aligned}$$
(102)

with solution:

$$\begin{aligned}&\left\langle K\right\rangle _{X}=\left( \frac{A}{\delta }f\left( X\right) \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }}\nonumber \\&\quad \times \left( \left\langle \begin{array}{c} 1-\frac{\kappa }{d^{2}}\int \left( K_{2}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\times \\ \times \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{array} \right\rangle \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\ \end{aligned}$$
(103)

This expression for \(\left\langle K\right\rangle _{X}\) can then be used in (100) to identify \(f\left( X\right) \):

$$\begin{aligned}&\left( f\left( X\right) \right) ^{1+\gamma } \nonumber \\&\quad =\frac{\kappa }{d^{2}} \left\langle \int \left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma } }\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) f\left( X_{2}\right) \right. \nonumber \\&\qquad \left. \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X_{2}-X_{3}\right| }{d}\right) \right) \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}\right\rangle \nonumber \\ \end{aligned}$$
(104)

and the expression for P follows from (99). We can now solve the defining Eqs. (103) and (104) by using a Green function approximation.

Expression of (103) using Green functions At the lowest order of perturbation theory, (104) can be rewritten using the Green functions associated to \(\left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\). Expression (104) becomes:

$$\begin{aligned}&\left( f\left( X\right) \right) ^{1+\gamma } \nonumber \\&\quad \simeq \frac{\kappa }{d^{2}} \int \left\{ G\left( \left( K_{3},X_{3},\theta \right) ,\left( K_{2},X_{2},\theta \right) \right) G\left( \left( K_{2},X_{2},\theta \right) ,\left( K_{3},X_{3},\theta \right) \right) \right. \nonumber \\&\qquad +\left. G\left( \left( K_{3},X_{3},\theta \right) ,\left( K_{3},X_{3},\theta \right) \right) G\left( \left( K_{2},X_{2},\theta \right) ,\left( K_{2},X_{2},\theta \right) \right) \right\} \nonumber \\&\qquad \times \left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) f\left( X_{2}\right) \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+ \frac{\left| X_{2}-X_{3}\right| }{d}\right) \right) \hbox {d}Z_{2}\hbox {d}Z_{3}\nonumber \\ \end{aligned}$$
(105)

We derive the Green functions in appendix 5. For \(\theta ^{\prime }\simeq \theta \), the Green functions \(G\left( \left( K_{3},X_{3}\right) ,\left( K_{2},X_{2}\right) \right) \) are those of harmonic oscillators for a propagation time of order \(\vartheta ^{2}\). Moreover, and discarding the \( \theta \) dependency, we have \(G\left( \left( K_{3},X_{2}\right) ,\left( K_{2},X_{2}\right) \right) \simeq 0\) and \(G\left( \left( K_{2},X_{2}\right) ,\left( K_{3},X_{3}\right) \right) \simeq 0\) for \(\theta ^{\prime }\simeq \theta \). We thus write (105):

$$\begin{aligned}&\left( f\left( X\right) \right) ^{1+\gamma } \nonumber \\&\quad \simeq \frac{\kappa }{d^{2}} \int G\left( \left( K_{3},X_{3}\right) ,\left( K_{3},X_{3}\right) \right) G\left( \left( K_{2},X_{2}\right) ,\left( K_{2},X_{2}\right) \right) \left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left( K_{3}\right) ^{-\frac{ \alpha }{1+\gamma }} \nonumber \\&\qquad \times f\left( X_{3}\right) f\left( X_{2}\right) \exp \left( -\left( \frac{ \left| X-X_{2}\right| }{d}+\frac{\left| X_{2}-X_{3}\right| }{ d}\right) \right) \hbox {d}Z_{2}\hbox {d}Z_{3} \end{aligned}$$
(106)

We also assume that the dynamics for K is faster than the dynamics for X , so that the Green functions have the form:

$$\begin{aligned} G\left( \left( K_{2},X_{3}\right) ,\left( K_{2},X_{2}\right) \right) \simeq G_{X_{2}}\left( K_{3},K_{2}\right) G\left( X_{2},X_{2}\right) \end{aligned}$$

This form will be justified by the formulas in appendix 5. As a consequence, the integrals over \(K_{2}\) and \(K_{3}\) in (106):

$$\begin{aligned} \int G_{X_{2}}\left( K_{2},K_{2}\right) \left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma } }G_{X_{3}}\left( K_{3},K_{2}\right) dK_{2}dK_{3} \end{aligned}$$

compute the average of \(\left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}\) for states such that \( X=X_{2}\) and \(X=X_{3}\), respectively. We write \(\left\langle K^{\frac{\alpha \gamma }{1+\gamma }}\right\rangle _{X_{2}}\) and \(\left\langle K^{-\frac{ \alpha }{1+\gamma }}\right\rangle _{X_{3}}\) these averages. Using that \( \left\langle K^{\frac{\alpha \gamma }{1+\gamma }}\right\rangle _{X_{2}}\simeq \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{ 1+\gamma }}\) and \(\left\langle K^{-\frac{\alpha }{1+\gamma }}\right\rangle _{X_{3}}\simeq \left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{ 1+\gamma }}\), it implies that:

$$\begin{aligned}&\int G\left( \left( K_{3},X_{3}\right) ,\left( K_{3},X_{3}\right) \right) G\left( \left( K_{2},X_{2}\right) ,\left( K_{2},X_{2}\right) \right) \left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left( K_{3}\right) ^{-\frac{ \alpha }{1+\gamma }}dK_{2}dK_{3} \nonumber \\&\quad \simeq G\left( X_{2},X_{3}\right) G\left( X_{3},X_{2}\right) \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{1+\gamma }} \end{aligned}$$
(107)

The average level of capital \(\left\langle K\right\rangle _{X}\) is computed below by identification.

As a consequence of (107), the expression (106) writes:

$$\begin{aligned} \left( f\left( X\right) \right) ^{1+\gamma }\simeq & {} \frac{\kappa }{d^{2}} \int G\left( X_{2},X_{2}\right) G\left( X_{2},X_{2}\right) \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{1+\gamma }} \nonumber \\&\times f\left( X_{3}\right) f\left( X_{2}\right) \exp \left( -\left( \frac{ \left| X-X_{2}\right| }{d}+\frac{\left| X_{2}-X_{3}\right| }{ d}\right) \right) \hbox {d}X_{2}\hbox {d}X_{3}\nonumber \\ \end{aligned}$$
(108)

The potential for the X-part of the action is harmonic with low frequency. We can thus assume that the integral in (108) is distributed around \( X_{2}-X_{3}\simeq 0\) and replace \(\exp \left( -\left( \frac{\left| X_{2}-X_{3}\right| }{d}\right) \right) \) by 2d times a delta function. As a consequence, (108) simplifies as:

$$\begin{aligned} \left( f\left( X\right) \right) ^{1+\gamma }\simeq \frac{2\kappa }{d}\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \left( \gamma -1\right) }{ 1+\gamma }}\left( f\left( X_{2}\right) \right) ^{2}G^{2}\left( X_{2},X_{2}\right) \exp \left( -\left( \frac{\left| X-X_{2}\right| }{ d}\right) \right) \hbox {d}X_{2} \end{aligned}$$
(109)

Given the low frequency describing the X-part of the action, the positions in the exchange space can be considered homogeneously spread on \(\left[ -1,1 \right] \), the probability \(G\left( X_{2},X_{2}\right) \) can be replaced by a constant density, here \(\frac{1}{2}\) in first approximation, to normalise the probability of the interval \(\left[ -1,1\right] \) to 1.Footnote 8 As a consequence, we have:

$$\begin{aligned} \left( f\left( X\right) \right) ^{1+\gamma }\simeq \frac{\kappa }{2d}\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \left( \gamma -1\right) }{ 1+\gamma }}\left( f\left( X_{2}\right) \right) ^{2}\exp \left( -\frac{ \left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2} \end{aligned}$$
(110)

Then, using (103), we can express \(\left\langle K\right\rangle _{X}\) as a function of X:

$$\begin{aligned}&\left\langle K\right\rangle _{X}=\left( \frac{A}{\delta }f\left( X\right) \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }}\nonumber \\&\quad \times \left( \begin{array}{c} 1-\frac{\kappa }{d^{2}}\int \left( K_{2}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\exp \left( -\left( \frac{\left| X-X_{2}\right| +\left| X-X_{3}\right| }{ d}\right) \right) \\ \times \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}X_{2}\hbox {d}X_{3} \end{array} \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\&\quad \simeq \left( \frac{A}{\delta }f\left( X\right) \right) ^{\frac{1+\gamma }{ 1+\gamma \left( 1-\alpha \right) }}\left( 1-\frac{\kappa }{2d^{2}}\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{1+\gamma } }\left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{1+\gamma }}\right. \nonumber \\&\qquad \times \left. f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{\left| X-X_{3}\right| }{d}\right) \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) \end{aligned}$$
(111)

for \(\gamma<<1\). The factor \(\frac{1}{4}\) normalises the probability of each interval \(\left[ -1,1\right] \) to 1.

Solving (111)

We can solve (111) to find the average capital \(\left\langle K\right\rangle _{X}\). To do so, we postulate the following first approximation for \(\left\langle K\right\rangle _{X}\):

$$\begin{aligned} \left\langle K\right\rangle _{X}= & {} \left( \frac{A}{\delta }f\left( X\right) \left( 1-h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( - \frac{\left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{X}{d}}{ \exp \left( \frac{1}{d}\right) }\right) \right) \right) ^{\frac{1+\gamma }{ 1+\gamma \left( 1-\alpha \right) }} \nonumber \\\simeq & {} \left( \frac{A}{\delta }f\left( X\right) \left( 1-h\exp \left( - \frac{\left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{X}{d}}{ \exp \left( \frac{1}{d}\right) }\right) \right) \right) ^{\frac{1+\gamma }{ 1+\gamma \left( 1-\alpha \right) }} \end{aligned}$$
(112)

with h to be determined. In (111) we can, for the sake of simplicity, replace \(\left\langle K\right\rangle _{X}\) in the integrals by its average over X, written \(\overline{\left\langle K\right\rangle }_{X}\) ( \(h<\exp \left( \frac{1}{2d}\right) \)). It is approximatively equal to:

$$\begin{aligned} \overline{\left\langle K\right\rangle }_{X}=\left( \frac{A}{\delta }f\left( X\right) \left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d} \right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }} \end{aligned}$$
(113)

Equation (113), along with (110), yields the following equation for \(f\left( X\right) \):

$$\begin{aligned}&\left( f\left( X\right) \right) ^{1+\gamma } \nonumber \\&\quad \simeq \frac{\kappa }{2d}\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \left( \gamma -1\right) }{ 1+\gamma }}\left( f\left( X_{2}\right) \right) ^{2}\exp \left( -\frac{ \left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2} \nonumber \\&\quad \simeq \frac{\kappa }{2d}\left( \frac{A}{\delta }\right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }}\int \left( \left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\&\qquad \times \left( f\left( X_{2}\right) \right) ^{2+\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }}\exp \left( -\frac{\left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2} \end{aligned}$$
(114)

We look for a solution of (114) of the form:

$$\begin{aligned} f\left( X\right) =D\exp \left( -c\frac{\left| X\right| }{d}\right) \end{aligned}$$
(115)

and Eq. (114) becomes for \(\gamma<<1\) :

$$\begin{aligned}&\exp \left( -\left( 1+\gamma \right) c\frac{\left| X\right| }{d} \right) \nonumber \\&\quad \simeq \frac{\kappa }{2d}\left( \frac{A}{\delta }\right) ^{\frac{ \alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }}D^{\frac{ \left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\&\qquad \times \left( \left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{ 2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\&\qquad \times \int \exp \left( -c\frac{\left( 1+\gamma \right) \left( 2-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }\frac{\left| X_{2}\right| }{d}\right) \exp \left( -\frac{\left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2} \end{aligned}$$
(116)

The integral in (116) can be estimated as (\(X>0\)):

$$\begin{aligned}&\int _{-1}^{0}\exp \left( \frac{2aX_{2}}{d}\right) \exp \left( -\frac{ \left( X-X_{2}\right) }{d}\right) +\int _{0}^{X}\exp \left( -\frac{2aX_{2}}{d} \right) \exp \left( -\frac{\left( X-X_{2}\right) }{d}\right) \\&\qquad +\int _{X}^{1}\exp \left( -\frac{2aX_{2}}{d}\right) \exp \left( \frac{ \left( X-X_{2}\right) }{d}\right) \\&\quad = d\left( \frac{\exp \left( -\frac{X}{d}\right) \left( 1-\exp \left( -\frac{ \left( 2a+1\right) }{d}\right) \right) }{2a+1}\right. \\&\left. \qquad +\left( \frac{\exp \left( - \frac{2aX}{d}\right) -\exp \left( -\frac{X}{d}\right) }{1-2a}\right) +\frac{ \exp \left( -\frac{2aX}{d}\right) -\exp \left( \frac{X-\left( 2a+1\right) }{d }\right) }{2a+1}\right) \\&\quad \simeq d\frac{4a\exp \left( -\frac{X}{d}\right) -2\exp \left( -\frac{2aX}{d }\right) }{4a^{2}-1} \end{aligned}$$

and the identification of \(f\left( X\right) \) in (116) becomes:

$$\begin{aligned}&\exp \left( -\left( 1+\gamma \right) c\frac{\left| X\right| }{d} \right) \nonumber \\&\quad =\frac{\kappa }{2d}\left( \frac{A}{\delta }\right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }}D^{\frac{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\&\qquad \times \left( \left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{ 2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }} \nonumber \\&\qquad \times \left( \frac{4a\exp \left( -\frac{X}{d}\right) -2\exp \left( -2a \frac{X}{d}\right) }{4a^{2}-1}\right) \end{aligned}$$
(117)

We will justify below that \(2\exp \left( -2a\frac{X}{d}\right) \) can be neglected. In this case, for \(\gamma<<1\), one has:

$$\begin{aligned} c\simeq & {} \frac{1}{1+\gamma } \\ 2a= & {} \frac{\left( 1+\gamma \right) \left( 2-\alpha \right) }{ 1+\gamma \left( 1-\alpha \right) }\frac{1}{1+\gamma } \\= & {} \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }>1 \end{aligned}$$

The fact that 2a is greater than 1 justifies our hypothesis to neglect \( 2\exp \left( -2a\frac{X}{d}\right) \) and leads to identifying the postulated constant D in (115) by writing (117) as:

$$\begin{aligned}&\frac{\kappa }{2}\left( \frac{A}{\delta }\right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }}D^{\frac{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }}\frac{\frac{2\left( 2-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }}{\left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1}\left( \left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d }\right) \right. \right. \\&\quad \left. \left. \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma \left( 1-\alpha \right) }}=1 \end{aligned}$$

whose solution is:

$$\begin{aligned} D\simeq & {} \left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right. \nonumber \\&\times \left. \left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{1}{2d} \right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1 }{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha \left( 1-\gamma \right) }{1+\gamma \left( 1-\alpha \right) }}\right) ^{\frac{1+\gamma \left( 1-\alpha \right) }{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }} \end{aligned}$$
(118)

We ultimately rewrite (115) as:

$$\begin{aligned} f\left( X\right)= & {} D\exp \left( -\frac{\left| X\right| }{\left( 1+\gamma \right) d}\right) \nonumber \\= & {} \left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha }{\left( 1-\alpha \right) \left( \gamma +1\right) }} \nonumber \\&\times \left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) ^{\frac{1+\gamma \left( 1-\alpha \right) }{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }}\exp \left( - \frac{\left| X\right| }{d\left( 1+\gamma \right) }\right) \nonumber \\ \end{aligned}$$
(119)

The full form of \(f\left( X\right) \) and \(\left\langle K\right\rangle _{X}\) is obtained by computing the constant h. It is found by the identification of (112) and (111), which yields:

$$\begin{aligned}&h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( -\frac{ \left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{X}{d}}{\exp \left( \frac{1}{d}\right) }\right) =\frac{\kappa }{4d^{2}}\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{1+\gamma } }\left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma } \nonumber \\&\qquad \times \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) \hbox {d}X_{2}\hbox {d}X_{3} \end{aligned}$$
(120)

Considering \(\gamma<<1\), we compute the RHS of (120) as:

$$\begin{aligned}&\frac{\kappa }{4d^{2}}\int \left\langle K\right\rangle _{X_{2}}^{\frac{ \alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3}}^{-\frac{ \alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+ \frac{\left| X-X_{3}\right| }{d}\right) \right) \hbox {d}X_{2}\hbox {d}X_{3} \\&\quad =\frac{\kappa }{4d^{2}}\left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{1}{ 2d}\right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( - \frac{1}{2d}\right) }{2}\right) \right) \right) ^{-\frac{\alpha \left( 1-\gamma \right) }{1+\gamma \left( 1-\alpha \right) }} \\&\qquad \times \int \left( f\left( X_{3}\right) \right) ^{\frac{\left( 1-\alpha \right) \left( 1+\gamma \right) }{1+\gamma \left( 1-\alpha \right) }}\left( f\left( X_{2}\right) \right) ^{-\gamma \frac{\left( 1-\alpha \right) \left( 1+\gamma \right) }{1+\gamma \left( 1-\alpha \right) }}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{\left| X-X_{3}\right| }{d}\right) \right) \hbox {d}X_{2}\hbox {d}X_{3} \end{aligned}$$

Using (119), this expression writes in first approximation:

$$\begin{aligned}&\frac{1}{4d^{2}}\left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) \nonumber \\&\qquad \times \int \exp \left( -\frac{\left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }\frac{\left| X_{3}\right| }{d}\right) \exp \left( -\gamma \frac{\left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) } \frac{\left| X_{2}\right| }{d}\right) \nonumber \\&\qquad \exp \left( -\left( \frac{ \left| X-X_{2}\right| }{d}+\frac{\left| X-X_{3}\right| }{d} \right) \right) \nonumber \\&\quad \simeq \frac{1}{4d^{2}}\left( \frac{1+\gamma \left( 1-\alpha \right) }{ \left( 2-\alpha \right) }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) \nonumber \\&\qquad \times \int \exp \left( -\frac{\left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }\frac{\left| X_{3}\right| }{d}\right) \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{\left| X-X_{3}\right| }{d}\right) \right) \nonumber \\&\quad = \frac{1}{2d}\frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) } \right) ^{2}-1\right) \nonumber \\&\qquad \times \int \exp \left( -\frac{\left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }\frac{\left| X_{3}\right| }{d}\right) \exp \left( - \frac{\left| X-X_{3}\right| }{d}\right) \left( 1-\frac{\cosh \left( \frac{X}{d}\right) }{\exp \left( \frac{1}{d}\right) }\right) \nonumber \\ \end{aligned}$$
(121)

where the equality:

$$\begin{aligned} \int \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}\right) \right) =2d\left( 1-\frac{\cosh \left( \frac{X}{d}\right) }{\exp \left( \frac{1}{d}\right) }\right) \end{aligned}$$
(122)

has been used. We then find h by computing the integral arising in (121) for \(X>0\), the general case being obtained by replacing X with \( \left| X\right| \).

$$\begin{aligned}&\int \exp \left( -\frac{\left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }\frac{\left| X_{3}\right| }{d}\right) \exp \left( -\frac{ \left| X-X_{3}\right| }{d}\right) \hbox {d}X_{3} \\&\quad \simeq \int \exp \left( -\left( 1-\alpha \right) \frac{\left| X_{3}\right| }{d}\right) \exp \left( -\frac{\left| X-X_{3}\right| }{d}\right) \hbox {d}X_{3} \\&\qquad \int _{-1}^{0}\exp \left( \left( 1-\alpha \right) \frac{u}{d}-\frac{X-u}{d} \right) du+\int _{X}^{1}\exp \left( -\left( 1-\alpha \right) \frac{u}{d}+ \frac{X-u}{d}\right) du \\&\qquad +\int _{0}^{X}\exp \left( -\left( 1-\alpha \right) \frac{u}{d}-\frac{X-u}{d} \right) du \\&\quad = d\left( \frac{e^{\alpha \frac{X}{d}}-1}{\alpha }e^{-\frac{X}{d}}+\frac{e^{ \frac{X+\alpha -2}{d}}-e^{\frac{X}{d}\left( \alpha -1\right) }}{\alpha -2} -e^{-\frac{X}{d}}\frac{1-e^{\frac{\alpha -2}{d}}}{\alpha -2}\right) \end{aligned}$$

The above equation can be approximated at the zeroth order in \(\alpha \):

$$\begin{aligned}&\int \exp \left( -\frac{\left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }\frac{\left| X_{3}\right| }{d}\right) \exp \left( -\frac{ \left| X-X_{3}\right| }{d}\right) \nonumber \\&\quad \simeq d\left( \frac{1}{2}e^{-\frac{X}{d}}-\frac{1}{2}e^{\frac{ X-2}{d}}-e^{-\frac{X}{d}}\left( \frac{1}{2}e^{-\frac{2}{d}}-\frac{1}{2} \right) +\frac{X}{d}e^{-\frac{X}{d}}\right) \nonumber \\&\quad \simeq d\left( 1+\frac{X}{d}-\frac{1}{2}e^{\frac{2X-2}{d}}\right) e^{- \frac{X}{d}} \end{aligned}$$
(123)

The correction terms of order \(\alpha \) are negligible for d close to 1. Moreover, for \(d\simeq 1\), formula (123) is equal to \(d\left( 1+\frac{X }{d}\right) e^{-\frac{X}{d}}\) in first approximation. Restoring the absolute value \(\left| X\right| \), inserting (123) in (121) and using (120) directly yields the value of h, for \(\gamma<<1\) and d close to 1:

$$\begin{aligned} h\simeq \left( \frac{1+\gamma \left( 1-\alpha \right) }{2\left( 2-\alpha \right) }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) } \right) ^{2}-1\right) \right) \end{aligned}$$
(124)

Once \(\left\langle K\right\rangle _{X}\) has been found, the price can be rewritten as a function of \(\left\langle K\right\rangle _{X}\) using (112):

$$\begin{aligned} P= & {} P\left( K,X\right) \\= & {} \left( K\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X\right) \\= & {} \left( \frac{K}{\left\langle K\right\rangle _{X}}\right) ^{-\frac{\alpha }{1+\gamma }}\frac{f\left( X\right) }{\left( \frac{A}{\delta }f\left( X\right) \left( 1-h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( -\frac{\left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{ X}{d}}{\exp \left( \frac{1}{d}\right) }\right) \right) \right) ^{\frac{ \alpha }{1+\gamma \left( 1-\alpha \right) }}} \\= & {} \left( \frac{K}{\left\langle K\right\rangle _{X}}\right) ^{-\frac{\alpha }{1+\gamma }}\frac{\left( f\left( X\right) \right) ^{\frac{\left( 1+\gamma \right) \left( 1-\alpha \right) }{1+\gamma \left( 1-\alpha \right) }}}{ \left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{\left| X\right| }{d} \right) \exp \left( -\frac{\left| X\right| }{d}\right) \left( 1- \frac{\cosh \frac{X}{d}}{\exp \left( \frac{1}{d}\right) }\right) \right) \right) ^{\frac{\alpha }{1+\gamma \left( 1-\alpha \right) }}} \end{aligned}$$

and thus, given (112):

$$\begin{aligned} P\left( K,X\right)= & {} \frac{\left( \frac{1+\gamma \left( 1-\alpha \right) }{ \left( 2-\alpha \right) \kappa }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) ^{\frac{1}{1-\gamma } }\exp \left( -\frac{\left( 1-\alpha \right) \left| X\right| }{ d\left( 1+\gamma \left( 1-\alpha \right) \right) }\right) }{\left( \frac{K}{ \left\langle K\right\rangle _{X}}\right) ^{\frac{\alpha }{1+\gamma }}\left( \left( \frac{1-h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( -\frac{\left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{ X}{d}}{\exp \left( \frac{1}{d}\right) }\right) }{1-h\left( 1+\frac{1}{2d} \right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1 }{2d}\right) }{2}\right) }\right) \right) ^{\frac{\alpha }{1+\gamma \left( 1-\alpha \right) }}} \nonumber \\\simeq & {} \frac{\left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) ^{\frac{1}{1-\gamma }}}{ \left( \frac{K}{\left\langle K\right\rangle _{X}}\right) ^{\frac{\alpha }{ 1+\gamma }}}\exp \left( -\frac{\left( 1-\alpha \right) \left| X\right| }{d\left( 1+\gamma \left( 1-\alpha \right) \right) }\right) \nonumber \\ \end{aligned}$$
(125)

Ultimately, the previous computations imply the following form for the potential:

$$\begin{aligned} \delta ^{2}\left( K-\left\langle K\right\rangle _{X}\right) ^{2} \end{aligned}$$

which justifies the assumption of harmonic oscillations.

Determination of prices and average capital as functions of \(\Psi \), expression of the potential. Corrections of order \( \left( \frac{A}{\bar{A}}\right) ^{2}\)

Having found \(\left\langle K\right\rangle _{X}\), the price \(P\left( K,X\right) \) and the potential for \(\bar{A}\rightarrow \infty \), we can now consider the corrections due to \(\frac{A}{\bar{A}}\):

\(\bar{A}>>A\): once again we consider fields of the form: \(\Psi \left( K,P,X,\theta \right) \rightarrow \delta \left( P-P\left( K,X,\theta \right) \right) \Psi \left( K,X,\theta \right) \).

The factor arising in the potential term of expression (20) is now:

$$\begin{aligned}&\left( \delta K-APK^{\alpha }\left( 1-\frac{\kappa }{d^{2}}\int \frac{ P_{3}\exp \left( -\left( \left| X-X_{2}\right| +\left| X-X_{3}\right| \right) \right) }{P_{2}^{\gamma }}\right. \right. \nonumber \\&\left. \left. \quad \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\right) \hbox {d}Z_{2}\hbox {d}Z_{3}\right) ^{2} \nonumber \\&\qquad +\bar{A}^{2}\left( P^{1+\gamma }K^{\alpha }-\frac{\kappa }{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\left( \left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| \right) \right) \right. \nonumber \\&\left. \qquad \left| \Psi \left( K_{2},X_{2},\theta \right) \right| ^{2}\left| \Psi \left( K_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}\right) ^{2} \nonumber \\&\quad =\left( \delta K-APK^{\alpha }U\right) ^{2}+\bar{A}^{2}\left( P^{1+\gamma }K^{\alpha }-V\right) ^{2} \end{aligned}$$
(126)

with:

$$\begin{aligned} U\left( X\right)= & {} 1-U_{1} \nonumber \\= & {} \left( 1-\frac{\kappa }{d^{2}}\int \frac{P_{3}\exp \left( -\left( \left| X-X_{2}\right| +\left| X-X_{3}\right| \right) \right) }{P_{2}^{\gamma }}\left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}\right) \nonumber \\ V\left( X\right)= & {} \frac{\kappa }{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\left( \left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| \right) \right) \left| \Psi \left( Z_{2},\theta \right) \right| ^{2}\left| \Psi \left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}\nonumber \\ \end{aligned}$$
(127)

As for other quantities, we replace \(U\left( X\right) \) and \(V\left( X\right) \) in (126) by their expectations \(\left\langle U\left( X\right) \right\rangle \) and \(\left\langle V\left( X\right) \right\rangle \). The value of \(\left\langle U\left( X\right) \right\rangle \) will be needed later and is given by identifying (112) and (111):

$$\begin{aligned} \left\langle U\left( X\right) \right\rangle= & {} 1-h\exp \left( -\frac{ \left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{X}{d}}{\exp \left( \frac{1}{d}\right) }\right) \nonumber \\\simeq & {} 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d} \right) }{2}\right) \end{aligned}$$
(128)

In the following, for the sake of simplicity in notations, U and V will stand for \(\left\langle U\right\rangle \) and \(\left\langle V\right\rangle \).

The price \(P\left( K,X\right) \) is found through the minimisation of the potential (126). This yields the condition:

$$\begin{aligned} -UA\left( \delta K-APK^{\alpha }U\right) +\left( 1+\gamma \right) P^{\gamma } \bar{A}^{2}\left( P^{1+\gamma }K^{\alpha }-V\right) =0 \end{aligned}$$
(129)

At the lowest order approximation of (129) in \(\gamma \) is:

$$\begin{aligned} -AU\left( \delta K-APK^{\alpha }U\right) +\bar{A}^{2}\left( P^{1+\gamma }K^{\alpha }-V\right) \simeq 0 \end{aligned}$$
(130)

We now use (130) to compute \(P^{1+\gamma }K^{\alpha }\) at first order in \(\frac{A^{2}}{\bar{A}^{2}}\):

$$\begin{aligned} P^{1+\gamma }K^{\alpha }= & {} \frac{\bar{A}^{2}}{A^{2}U^{2}+\bar{A}^{2}}V+ \frac{AU}{A^{2}U^{2}+\bar{A}^{2}}\delta K \\= & {} V+\frac{AU}{A^{2}U^{2}+\bar{A}^{2}}\left( \delta K-AUV\right) \end{aligned}$$

At this order, this also rewrites:

$$\begin{aligned} P^{1+\gamma }K^{\alpha }= & {} V+\frac{AU}{A^{2}U^{2}+\bar{A}^{2}}\left( \delta K-AUP^{1+\gamma }K^{\alpha }\right) \nonumber \\\simeq & {} V+\frac{AU}{A^{2}U^{2}+\bar{A}^{2}}\left( \delta K-AUPK^{\alpha }\right) \end{aligned}$$
(131)

For this value, the factor (126) becomes:

$$\begin{aligned}&\left( \delta K-APK^{\alpha }U\right) ^{2}+\bar{A}^{2}\left( P^{1+\gamma }K^{\alpha }-V\right) ^{2} \nonumber \\&\quad \simeq \left( \delta K-APK^{\alpha }U\right) ^{2}+\bar{A} ^{2}A^{2}U^{2}\left( \frac{\delta K-APUK^{\alpha }}{A^{2}U^{2}+\bar{A}^{2}} \right) ^{2} \nonumber \\&\quad \simeq \delta ^{2}\left( 1+\frac{A^{2}U^{2}}{A^{2}U^{2}+\bar{A}^{2}} \right) \left( K-\frac{APK^{\alpha }U}{\delta }\right) ^{2} \end{aligned}$$
(132)

Equations (131) and (132) can be simplified in order to find \( P\left( K,X\right) \) and \(\left\langle K\right\rangle _{X}\). Since we are looking for first-order corrections in \(\frac{A^{2}}{\bar{A}^{2}}\), we can replace \(\delta K-AV\) in (131) by its lowest order approximation, that is, given our assumption \(\gamma<<1\):

$$\begin{aligned} \delta K-AV=\delta K-AP^{1+\gamma }K^{\alpha }\simeq \delta K-APK^{\alpha } \end{aligned}$$

To this order of approximation, we can also replace K by \(\left\langle K\right\rangle _{X}\). Using (102):

$$\begin{aligned} \delta \left\langle K\right\rangle _{X}=AP\left\langle K\right\rangle _{X}^{\alpha }U \end{aligned}$$
(133)

and (112):

$$\begin{aligned} \frac{\delta K-AUV}{V}\simeq A\frac{\delta \left\langle K\right\rangle _{X}-APU\left\langle K\right\rangle _{X}^{\alpha }}{AP\left\langle K\right\rangle _{X}^{\alpha }}=0 \end{aligned}$$

Equation (131) writes at the first order in \(\frac{A^{2}}{\bar{A}^{2}} \):

$$\begin{aligned} P^{1+\gamma }K^{\alpha }=V \end{aligned}$$
(134)

As a consequence of (132) and (134), the equations for \( P\left( K,X\right) \) and \(\left\langle K\right\rangle _{X}\) with respect to the case \(\bar{A}\rightarrow \infty \) are unchanged, as well as their solutions (112) and (125). The term (132) rewrites:

$$\begin{aligned} \delta ^{2}\left( 1+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}\right) \left( K-\frac{AUPK^{\alpha }}{\delta }\right) ^{2}\simeq \delta ^{2}\left( 1+\frac{A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) }\right) \left( K-\left\langle K\right\rangle _{X}\right) ^{2} \end{aligned}$$
(135)

Accounting for the normalisation factor

At this point, the normalisation factor \(2d\left( 1-\frac{\cosh \frac{ X_{i}\left( t\right) }{d}}{\exp \left( \frac{1}{d}\right) }\right) \) for the price index \(\hat{P}_{i}\left( t\right) \) that was skipped for the sake of simplicity can be introduced directly without modifying the main results. An inspection of (6) and (8) shows that it amounts to replacing inside the integrals in (97) and (98) a factor \( \frac{1}{d}\) by \(\left( 2d\left( 1-\frac{\cosh \frac{X_{2}}{d}}{\exp \left( \frac{1}{d}\right) }\right) \right) ^{-1}\) and \(\left( 2d\left( 1-\frac{ \cosh \frac{X}{d}}{\exp \left( \frac{1}{d}\right) }\right) \right) ^{-1}\), respectively. The introduction of these factors can be accounted for by keeping (97) and (98) unchanged, as well as the following computations, except for the integrals \(\int \exp \left( -\left( \frac{ \left| X_{2}-X_{3}\right| }{d}\right) \right) \hbox {d}X_{3}\) in (108), and \(\int \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d} \right) \right) \hbox {d}X_{2}\) in (122) that have to be replaced by \(d\delta \left( X_{2}-X_{3}\right) \) and d, respectively. These modifications merely affect the formula for \(\left\langle K\right\rangle _{X}\) and D that would then write:

$$\begin{aligned} \left\langle K\right\rangle _{X}= & {} \left( \frac{A}{\delta }f\left( X\right) \left( 1-h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( - \frac{\left| X\right| }{d}\right) \right) \right) ^{\frac{1+\gamma }{ 1+\gamma \left( 1-\alpha \right) }}\\ D\simeq & {} \left( 2\frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa }\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right. \\&\times \left. \left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{1}{2d} \right) \exp \left( -\frac{1}{2d}\right) \right) \right) ^{\frac{\alpha \left( 1-\gamma \right) }{1+\gamma \left( 1-\alpha \right) }}\right) ^{\frac{ 1+\gamma \left( 1-\alpha \right) }{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }} \end{aligned}$$

with h still given by (124). As explained in the text, this does not modify the general interpretation for \(\left\langle K\right\rangle _{X}\) and the other variables, since \(1-\frac{\cosh \frac{X}{d}}{\exp \left( \frac{1}{d }\right) }\) decreases and varies slowly over \(\left[ -1,1\right] \).

Appendix 4. Determination of prices and average capital as functions of \(\Psi \) for \(\rho \ne 0\)

For a non-trivial fundamental defined by \(\rho >0\) [see (92) and (93)], the contribution of the background field \(\Psi _{0}\) has to be added to the fluctuations. Previous computations are valid, but average values have now to be computed in the state \(\Psi _{0}+\Delta \Psi \). We first consider the sole state \(\Psi _{0}\left( K,P,X,\theta \right) \), in which we find the corresponding expression for the prices \(P\left( K,X\right) \). We then derive the equations for \(f\left( X\right) \) and \( \left\langle K\right\rangle _{X}\) and compute the form of the fundamental \( \Psi _{0}\) and the value of \(\rho \). Finally, we compute the correction due to \(\Psi \left( K,P,X,\theta \right) \) and find the effective action.

Equations for P, \(f\left( X\right) \) and \(\left\langle K\right\rangle _{X}\) in the state \(\Psi _{0}\)

Defining equation for \(\left\langle K\right\rangle _{X}\) In the state \(\Psi _{0}\left( K,P,X,\theta \right) \), the equations defining the potential P and the average capital \(\left\langle K\right\rangle _{X}\) are the same as in appendix 3. In the following, we proceed as in appendix 3. We first consider the case \(\bar{A}\rightarrow \infty \). We will ultimately include the first-order corrections in \(\frac{A^{2}}{\bar{A}^{2}}\) , since it amounts to modifying the potential (see 135).

To compute \(f\left( X\right) \), we first find the average of K, denoted \( \left\langle K\right\rangle _{X}\). It is computed in a state \(\rho \Psi _{0}\left( K,X\right) \). Given our order of approximations for \(\Psi _{0}\left( K,X\right) \), this amounts to replacing \(\kappa \) by \(\kappa \rho ^{4}\).

Here again we set a trial function for the price:

$$\begin{aligned} P=\left( K\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X\right) \end{aligned}$$
(136)

The equation for \(f\left( X\right) \) is the same as (104):

$$\begin{aligned}&f^{1+\gamma }\left( X\right) =\frac{\kappa \rho ^{4}}{d^{2}}\left\langle \int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\left( \left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| \right) \right) \right. \nonumber \\&\left. \quad \left| \Psi _{0}\left( Z_{2},\theta \right) \right| ^{2}\left| \Psi _{0}\left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3}\right\rangle \end{aligned}$$
(137)

Equation (103) for \(\left\langle K\right\rangle _{X}\) is still valid in first approximation:

$$\begin{aligned}&\left\langle K\right\rangle _{X}=\left( \frac{A}{\delta }f\left( X\right) \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }}\\&\quad \left\langle \begin{array}{c} 1-\frac{\kappa \rho ^{4}}{d^{2}}\int \left( K_{2}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left( K_{3}\right) ^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\\ \times \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}+\frac{ \left| X-X_{3}\right| }{d}\right) \right) \left| \Psi _{0}\left( Z_{2},\theta \right) \right| ^{2}\left| \Psi _{0}\left( Z_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{array} \right\rangle ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) }} \end{aligned}$$

For reasons similar to appendix 3, the state \(\Psi _{0}^{\dag }\left( K,X\right) \) is centred around \(\left\langle K\right\rangle _{X}\). The equation for \(f\left( X\right) \) is thus:

$$\begin{aligned} \left( f\left( X\right) \right) ^{1+\gamma }\simeq \frac{2\kappa \rho ^{4}}{d }\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f\left( X_{2}\right) \right) ^{2}G^{2}\left( X_{2},X_{2}\right) \exp \left( -\left( \frac{\left| X-X_{2}\right| }{ d}\right) \right) \hbox {d}X_{2} \end{aligned}$$

In the state \(\Psi _{0}\left( K,X\right) \), \(G\left( X_{2},X_{2}\right) \) simply denotes \(\int \left| \Psi _{0}\left( K,X\right) \right| ^{2}dK \). Moreover, in first approximation, the defining equation for the X-part of \(\Psi _{0}\left( K,X\right) \) is an oscillator with \(\kappa _{0}<<1\) . The distribution for X defined by \(G\left( X_{2},X_{2}\right) \) can thus be considered as uniform. Equations (110) and (111) for \( f\left( X\right) \) and K are similar to those obtained in appendix 3, up to a factor \(\rho ^{4}\), and write:

$$\begin{aligned}&\left( f\left( X\right) \right) ^{1+\gamma }\simeq \frac{\kappa \rho ^{4}}{2d }\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f\left( X_{2}\right) \right) ^{2}\exp \left( - \frac{\left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2} \end{aligned}$$
(138)
$$\begin{aligned}&\left\langle K\right\rangle _{X} \simeq \left( \frac{A}{\delta }f\left( X\right) \right) ^{\frac{1+\gamma }{1+\gamma \left( 1-\alpha \right) } }\left( 1-\frac{\kappa \rho ^{4}}{2d}\int \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\right. \nonumber \\&\qquad \qquad \,\times \left. \exp \left( -\left( \frac{ \left| X-X_{2}\right| }{d}+\frac{\left| X-X_{3}\right| }{d} \right) \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) \end{aligned}$$
(139)

Solving (139) As in appendix 3, the form of \(\left\langle K\right\rangle _{X}\) is postulated to solve (139):

$$\begin{aligned} \left\langle K\right\rangle _{X}=\left( \frac{A}{\delta }f\left( X\right) \left( 1-h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( - \frac{\left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{X}{d}}{ \exp \left( \frac{1}{d}\right) }\right) \right) \right) ^{\frac{1+\gamma }{ 1+\gamma \left( 1-\alpha \right) }} \end{aligned}$$
(140)

The resolution is thus similar to appendix 3, and we find:

$$\begin{aligned} f\left( X\right)= & {} D_{\rho }\exp \left( -\frac{\left| X\right| }{ 1+\gamma }\right) \nonumber \\= & {} \left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha }{\left( 1-\alpha \right) \left( \gamma +1\right) }} \nonumber \\&\times \left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa \rho ^{4}}\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) ^{\frac{1+\gamma \left( 1-\alpha \right) }{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) } }\exp \left( -\frac{\left| X\right| }{d\left( 1+\gamma \right) } \right) \nonumber \\ \end{aligned}$$
(141)

where:

$$\begin{aligned} D_{\rho }= & {} \left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa \rho ^{4}}\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right. \nonumber \\&\times \left. \left( \frac{A}{\delta }\left( 1-h\left( 1+\frac{1}{2d} \right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1 }{2d}\right) }{2}\right) \right) \right) ^{\frac{\alpha \left( 1-\gamma \right) }{1+\gamma \left( 1-\alpha \right) }}\right) ^{\frac{1+\gamma \left( 1-\alpha \right) }{\left( 1-\gamma ^{2}\right) \left( 1-\alpha \right) }} \end{aligned}$$
(142)

and:

$$\begin{aligned} h=\left( \frac{1+\gamma \left( 1-\alpha \right) }{2\left( 2-\alpha \right) } \left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) \end{aligned}$$
(143)

Using (136) yields the expression for the price:

$$\begin{aligned} P= & {} \frac{\left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa \rho ^{4}}\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) ^{\frac{1}{1-\gamma }}\exp \left( -\frac{\left( 1-\alpha \right) \left| X\right| }{d\left( 1+\gamma \left( 1-\alpha \right) \right) }\right) }{\left( \frac{K}{ \left\langle K\right\rangle _{X}}\right) ^{\frac{\alpha }{1+\gamma }}\left( \left( \frac{1-h\left( 1+\frac{\left| X\right| }{d}\right) \exp \left( -\frac{\left| X\right| }{d}\right) \left( 1-\frac{\cosh \frac{ X}{d}}{\exp \left( \frac{1}{d}\right) }\right) }{1-h\left( 1+\frac{1}{2d} \right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1 }{2d}\right) }{2}\right) }\right) \right) ^{\frac{\alpha }{1+\gamma \left( 1-\alpha \right) }}} \nonumber \\\simeq & {} \frac{\left( \frac{1+\gamma \left( 1-\alpha \right) }{\left( 2-\alpha \right) \kappa \rho ^{4}}\left( \left( \frac{2-\alpha }{1+\gamma \left( 1-\alpha \right) }\right) ^{2}-1\right) \right) ^{\frac{1}{1-\gamma }} }{\left( \frac{K}{\left\langle K\right\rangle _{X}}\right) ^{\frac{\alpha }{ 1+\gamma }}}\exp \left( -\frac{\left( 1-\alpha \right) \left| X\right| }{d\left( 1+\gamma \left( 1-\alpha \right) \right) }\right) \quad \end{aligned}$$
(144)

These results depend on \(\rho \). The next paragraph will derive the value of \(\rho \) and the expression of \(\Psi _{0}\).

Computation of \(\rho \) and \(\Psi _{0}\) The value of \(\rho \) and the form of \(\Psi _{0}\) can now be computed. As explained before, this phase is possible approximatively for:

$$\begin{aligned} \kappa _{1}^{2}-2\kappa _{0}^{\frac{1}{2}}\kappa _{2}>0 \end{aligned}$$

In this phase, \(\rho \) can be found approximatively by taking into account only the contribution in X:

$$\begin{aligned} \rho \simeq \frac{\kappa _{1}+\sqrt{\kappa _{1}^{2}-2\kappa _{0}^{\frac{1}{2} }\kappa _{2}}}{2\kappa _{2}} \end{aligned}$$

with \(\frac{\partial ^{2}S}{\partial \rho ^{2}}>0\). A more precise value of \( \rho \) is found by writing the equation for the state \(\Psi _{0}\left( K,X\right) \). Discarding the dependency in K to shorten the expressions, we have for the X-part of the action:

$$\begin{aligned} S_{2}\left( \Psi \left( X\right) \right)= & {} \int \Psi ^{\dag }\left( Z,\theta \right) \left( -\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}\right) \Psi \left( Z,\theta \right) +\frac{\kappa _{0}}{2\sigma _{X}^{2}}\Psi ^{\dag }\left( X\right) \left( X-\left\langle X\right\rangle \right) ^{2}\Psi \left( X\right) \nonumber \\&+\int \left( \Psi ^{\dag }\left( X\right) \Psi \left( X\right) \right) V_{1}\left( \left| X-Y\right| \right) \left( \Psi ^{\dag }\left( Y\right) \Psi \left( Y\right) \right) \nonumber \\&+\int \left( \Psi ^{\dag }\left( X\right) \Psi \left( X\right) \right) V_{2}\left( \left| X-Y\right| ,\left| X-Z\right| ,\left| Y-Z\right| \right) \left( \Psi ^{\dag }\left( Y\right) \Psi \left( Y\right) \right) \nonumber \\&\left( \Psi ^{\dag }\left( Z\right) \Psi \left( Z\right) \right) \end{aligned}$$
(145)

where:

$$\begin{aligned} V_{1}\left( \left| X-Y\right| \right)= & {} -\frac{\kappa _{1}}{2}\frac{ KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{ \left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}} \\ V_{2}\left( \left| X-Y\right| ,\left| X-Z\right| ,\left| Y-Z\right| \right)= & {} \frac{\kappa _{2}}{3}\exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \end{aligned}$$

The expression for \(\frac{\delta S_{2}\left( \Psi \left( X\right) \right) }{ \delta \Psi ^{\dag }\left( X\right) }\):

$$\begin{aligned}&-\frac{\sigma _{X}^{2}}{2}\nabla ^{2}\Psi \left( X\right) +\frac{\kappa _{0}}{2\sigma _{X}^{2}}\left( X-\left\langle X\right\rangle \right) ^{2}\Psi \left( X\right) +2V_{1}\left( \left| X-Y\right| \right) \left| \Psi \left( Y\right) \right| ^{2}\Psi \left( X\right) \nonumber \\&\quad +3V_{2}\left( \left| X-Y\right| ,\left| X-Z\right| ,\left| Y-Z\right| \right) \left| \Psi \left( Y\right) \right| ^{2}\left| \Psi \left( Z\right) \right| ^{2}\Psi \left( X\right) \end{aligned}$$
(146)

can be approximated by computing the potentials \(V_{1}\left( \left| X-Y\right| \right) \) and \(V_{2}\left( \left| X-Y\right| ,\right. \left. \left| X-Z\right| ,\left| Y-Z\right| \right) \) in the fundamental. As a consequence:

$$\begin{aligned} \int \kappa _{1}\frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2}dYdK^{\prime } \end{aligned}$$

is replaced by:

$$\begin{aligned}&\int \kappa _{1}\frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi _{0}\left( K^{\prime },Y\right) \right| ^{2}dYdK^{\prime } \\&\quad \simeq \kappa _{1}\exp \left( -\chi _{1}\left| X\right| \right) \frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2} \\&\quad \simeq \kappa _{1}\exp \left( -\chi _{1}\left| X\right| \right) \rho ^{2} \end{aligned}$$

where we assumed that, in first approximation, \(\left| \Psi _{0}\left( K^{\prime },Y\right) \right| ^{2}\)is centred around \(Y=0\). It will be justified by the formula for \(\Psi _{0}\).

By the same token, the second part of the potential:

$$\begin{aligned} \kappa _{2}\exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K'',Z\right) \right| ^{2} \end{aligned}$$

is replaced by:

$$\begin{aligned}&\int \kappa _{2}\exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi _{0}\left( K,Y\right) \right| ^{2}\left| \Psi _{0}\left( K,Z\right) \right| ^{2}\hbox {d}Y\hbox {d}Z \\&\quad \simeq \kappa _{2}\exp \left( -2\chi _{2}\left| X\right| \right) \rho ^{4} \end{aligned}$$

Moreover, in first approximation, for \(\chi _{1}<<1\), \(\chi _{2}<<1\), we can replace X with its expectation \(\left\langle X\right\rangle =0\). This leads to the overall expression for \(\frac{\delta S_{2}\left( \Psi \left( X\right) \right) }{\delta \Psi ^{\dag }\left( X\right) }\):

$$\begin{aligned}&-\frac{\sigma _{X}^{2}}{2}\nabla ^{2}\Psi \left( X\right) +\frac{\kappa _{0}}{2\sigma _{X}^{2}}X^{2}\Psi \left( X\right) \nonumber \\&\quad -\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}}\exp \left( -\chi _{1}\left| X\right| \right) \Psi \left( K,X\right) \rho ^{2}+\kappa _{2}\exp \left( -2\chi _{2}\left| X\right| \right) \Psi \left( K,X\right) \rho ^{4}+\alpha \Psi \left( X\right) \nonumber \\ \end{aligned}$$
(147)

Adding the K-part of the action to (147), the equation for the fundamental state at the lowest order in \(\chi _{1}\) and \(\chi _{2}\) is:

$$\begin{aligned} 0= & {} \left( -\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}-\frac{\sigma ^{2}}{2} \nabla _{K}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}\right) \Psi _{0}\left( K,X,\theta \right) \nonumber \\&+\left( \frac{1}{2\sigma ^{2}}\left( \delta ^{2}+\frac{\bar{A}^{2}A^{2}}{ \left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}\right) \left( K-\left\langle K\right\rangle _{X}\right) ^{2}\right. \nonumber \\&\left. +\frac{\kappa _{0}}{2\sigma _{X}^{2}} X^{2}+\left( \rho ^{4}\kappa _{2}-\rho ^{2}\kappa _{1}+\frac{1}{2\vartheta ^{2}}+\alpha \right) \right) \Psi _{0}\left( K,X,\theta \right) \end{aligned}$$
(148)

We will consider below the first-order corrections in \(\chi _{1}\)and \(\chi _{2}\) to this equation.

The Fourier transform of (148) in \(\theta \) shows that the fundamental does not depend on \(\theta \). We look for a fundamental eigenstate of the operator of the form \(N\Psi _{0}\left( X\right) \Psi _{0}^{\left( 2\right) }\left( K\right) \). We assume \(\theta <\Theta \) with \(\Theta>>1\), so that the integral over \(\theta \) exists.

The normalisation factor N ensures that \(N\Psi _{0}\left( X\right) \Psi _{0}^{\left( 2\right) }\left( K\right) \) has norm \(\rho ^{2}\). Then \(\Psi _{0}\left( X\right) \) and \(\Psi _{0}^{\left( 2\right) }\left( K\right) \) can be written as the fundamental states of oscillators with eigenstates:

$$\begin{aligned} \frac{\kappa _{0}^{\frac{1}{2}}}{2}\text { and }\frac{1}{2}\sqrt{\delta ^{2}+ \frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}} \end{aligned}$$

This leads to the relation:

$$\begin{aligned} 0=\alpha +\frac{1}{2\vartheta ^{2}}+\frac{1}{2}\kappa _{0}^{\frac{1}{2} }-\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}+\frac{\sqrt{\delta ^{2}+\frac{ \bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}{2} \end{aligned}$$

with:

$$\begin{aligned} U= & {} \left( 1-D^{1+\gamma }\int \exp \left( -\frac{\left| X_{3}\right| }{\left( 1+\gamma \right) }\right) \right. \\&\left. \exp \left( \frac{\gamma \left| X_{2}\right| }{\left( 1+\gamma \right) }\right) \exp \left( -\left( \left| X-X_{2}\right| +\left| X-X_{3}\right| \right) \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) \\\simeq & {} 1-h\left( 1+\frac{1}{2d}\right) \exp \left( -\frac{1}{2d}\right) \left( 1-\frac{\exp \left( -\frac{1}{2d}\right) }{2}\right) \end{aligned}$$

whose solution, with \(\frac{\partial ^{2}}{\partial \rho ^{2}}>0\), is:

$$\begin{aligned} \rho ^{2}=\frac{\kappa _{1}+\sqrt{\kappa _{1}^{2}-2\kappa _{2}\left( 2\alpha +\frac{1}{\vartheta ^{2}}+\sqrt{\kappa _{0}}+\sqrt{\delta ^{2}+\frac{\bar{A} ^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}\right) }}{2\kappa _{2} } \end{aligned}$$
(149)

and the eigenstate:

$$\begin{aligned} \Psi _{0}\left( K,X\right) =N\exp \left( -\frac{\kappa _{0}^{\frac{1}{2} }X^{2}}{2\sigma _{X}^{2}}\right) \exp \left( -\frac{\sqrt{\delta ^{2}+\frac{ \bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}\left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{2\sigma ^{2}}\right) \end{aligned}$$
(150)

where N is the normalisation factor:

$$\begin{aligned} N=\frac{\sqrt{\kappa _{0}^{\frac{1}{2}}\sqrt{\delta ^{2}+\frac{\bar{A} ^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}}{2\pi \sigma _{X}\sigma } \end{aligned}$$
(151)

which completes the computations for \(P\), \(\left\langle K\right\rangle _{X}\), the fundamental state and the potential.

We ultimately derive for later purpose the corrections to these results at the second order in \(\chi _{1}\) and \(\chi _{2}\). At this order, expanding the exponentials in \(\chi _{1}\) and \(\chi _{2}\) yields the potential part of the fundamental state’s equation (147):

$$\begin{aligned}&\frac{1}{2}\frac{\kappa _{0}}{\sigma _{X}^{2}}X^{2}-\rho ^{2}\kappa _{1}+\rho ^{4}\kappa _{2}+\kappa _{1}\chi _{1}\left| X\right| \frac{K }{\left\langle K\right\rangle _{X}}\rho ^{2}-2\kappa _{2}\chi _{2}\left| X\right| \rho ^{4}-\frac{\kappa _{1}}{2}\chi _{1}^{2}X^{2}\frac{K}{ \left\langle K\right\rangle _{X}}\rho ^{2}\nonumber \\&\qquad +2\kappa _{2}\chi _{2}^{2}X^{2}\rho ^{4} \nonumber \\&\quad =\frac{1}{2}\frac{\kappa _{0}}{\sigma _{X}^{2}}\left( X-\frac{sgn\left( X\right) \sigma _{X}^{2}}{\kappa _{0}}\left( -\chi _{1}\kappa _{1}\frac{K}{ \left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2}\nonumber \\&\qquad -\frac{\sigma _{X}^{2}}{2\kappa _{0}}\left( \chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2} \nonumber \\&\qquad +\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) X^{2}-\rho ^{2}\kappa _{1}+\rho ^{4}\kappa _{2} \nonumber \\&\quad \simeq \left( \frac{\kappa _{0}}{2\sigma _{X}^{2}}+\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) \right) \nonumber \\&\qquad \left( X-\frac{ sgn\left( X\right) \sigma _{X}^{2}}{\kappa _{0}}\left( -\chi _{1}\kappa _{1} \frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2} \nonumber \\&\qquad -\frac{\sigma _{X}^{2}}{2\kappa _{0}}\left( \chi _{1}\kappa _{1}\rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2}-\rho ^{2}\kappa _{1}+\rho ^{4}\kappa _{2} \end{aligned}$$
(152)

so that the fundamental equation becomes:

$$\begin{aligned}&0=\left( -\frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}-\frac{\sigma ^{2}}{2} \nabla _{K}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}\right) \Psi _{0}\left( K,X,\theta \right) \nonumber \\&\quad \qquad +\frac{\kappa _{0}}{2\sigma _{X}^{2}}\left( 1+\frac{2\sigma _{X}^{2}}{ \kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) \right) \nonumber \\&\quad \qquad \times \left( X-\frac{sgn\left( X\right) \sigma _{X}^{2}}{\kappa _{0}} \left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2}\Psi _{0}\left( K,X,\theta \right) \nonumber \\&\quad \qquad +\frac{1}{2\sigma ^{2}}\left( \left( \delta ^{2}+\frac{\bar{A}^{2}A^{2}}{ \left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}\right) \left( K-\left\langle K\right\rangle _{X}\right) ^{2}\right) \Psi _{0}\left( K,X,\theta \right) \nonumber \\&\quad \qquad +\left( \rho ^{4}\kappa _{2}-\rho ^{2}\kappa _{1}-\frac{\sigma _{X}^{2}}{ 2\kappa _{0}}\left( \chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2}+\frac{1}{ 2\vartheta ^{2}}+\alpha \right) \Psi _{0}\left( K,X,\theta \right) \nonumber \\ \end{aligned}$$
(153)

Equation (153) shows that the operators in K and X are intertwined. Yet, in first approximation in parameters \(\chi _{1}\) and \(\chi _{2}\), we can look for a fundamental state of the form \(N\Psi _{0}^{\left( 1\right) }\left( X\right) \Psi _{0}^{\left( 2\right) }\left( K\right) \). The expressions for \(\Psi _{0}^{\left( 1\right) }\left( X\right) \) and \(\Psi _{0}^{\left( 2\right) }\left( K\right) \) will depend perturbatively on K and X, which justifies the notations.

The operators in X and K are harmonic oscillators with frequencies:

$$\begin{aligned} \kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( - \frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X}} \rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }\text { \ and \ }\sqrt{ \delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2} }} \end{aligned}$$

The fundamental states of these operators have thus the form:

$$\begin{aligned} \Psi _{0}^{\left( 1\right) }\left( X\right)= & {} \exp \left( -\frac{\omega _{X}\left( X-\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( -\chi _{1}\kappa _{1} \frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2}}{2\sigma _{X}^{2}}\right) H\left( X\right) \nonumber \\&+\exp \left( -\frac{\omega _{X}\left( X+\frac{2\sigma _{X}^{2}}{\kappa _{0} }\left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2}}{2\sigma _{X}^{2}} \right) H\left( -X\right) \nonumber \\\equiv & {} \bar{\Psi }_{0}^{\left( 1\right) }\left( X-\frac{2\sigma _{X}^{2}}{ \kappa _{0}}\left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) H\left( X\right) \nonumber \\&+\bar{\Psi }_{0}^{\left( 1\right) }\left( X+\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X}} \rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) H\left( -X\right) \end{aligned}$$
(154)

with:

$$\begin{aligned} \omega _{X}= & {} \kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{ \kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) } \nonumber \\\simeq & {} \kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0} }\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) } \end{aligned}$$
(155)

and:

$$\begin{aligned} \Psi _{0}^{\left( 2\right) }\left( K\right) =\exp \left( -\frac{\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}} \left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{2\sigma ^{2}}\right) \end{aligned}$$
(156)

so that the total fundamental state in both variables writes:

$$\begin{aligned} \Psi _{0}\left( K,X\right)= & {} \rho N\left[ \bar{\Psi }_{0}^{\left( 1\right) }\left( X-\delta X\right) H\left( X\right) +\bar{\Psi }_{0}^{\left( 1\right) }\left( X+\delta X\right) H\left( -X\right) \right] \nonumber \\&\times \exp \left( -\frac{\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}\left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{2\sigma ^{2}}\right) \end{aligned}$$
(157)

where N is a normalisation factor:

$$\begin{aligned} N\simeq \frac{\sqrt{\kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{ \kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{ \left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}}{2\pi \sigma _{X}\sigma } \end{aligned}$$
(158)

for \(\delta X<<1\) and:

$$\begin{aligned} \delta X=\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( -\chi _{1}\kappa _{1} \frac{K}{\left\langle K\right\rangle _{X}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \end{aligned}$$
(159)

with \(\rho \) satisfying the condition:

$$\begin{aligned} 0\simeq & {} \left( \rho ^{4}\kappa _{2}-\rho ^{2}\kappa _{1}-\frac{\sigma _{X}^{2}}{2\kappa _{0}}\left( \chi _{1}\kappa _{1}\rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2}+\frac{1}{2\vartheta ^{2}}+\alpha \right) \nonumber \\&+\frac{1}{2}\kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{ \kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }+\frac{\sqrt{\delta ^{2}+\frac{\bar{A} ^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}{2} \nonumber \\ \end{aligned}$$
(160)

Contribution of \(\Delta \Psi \) We have computed the contribution of the fundamental \(\Psi _{0}\) to \( \left\langle K\right\rangle _{X}\). We must now find the correction to \( \left\langle K\right\rangle _{X}\) and P due to \(\Delta \Psi \) in the decomposition \(\Psi =\Psi _{0}+\Delta \Psi \). To do so, we compute the corrections to the potential in (20) due to \(\Delta \Psi \). We first consider the case of \(\Delta \Psi \) orthogonal to \(\Psi _{0}\) and ultimately add the contribution for \(\Delta \Psi \) proportional to \(\Psi _{0} \).

We can replace \(\Psi \left( K_{3},P_{3},X_{3},\theta \right) \ \)in the previous computations by \(\Psi _{0}\left( K_{3},P_{3},X_{3},\theta \right) +\Delta \Psi \left( K_{3},P_{3},X_{3},\theta \right) \) as follows.

For any quantity \(\left( K\right) ^{-\frac{\alpha }{1+\gamma }}\) and \( f\left( X\right) \), the expectation of the vector \(\left( \begin{array}{c} \left( K\right) ^{-\frac{\alpha }{1+\gamma }} \\ f\left( X\right) \end{array} \right) \) is defined by:

$$\begin{aligned}&\left( \Psi _{0}^{\dag }\left( K_{3},P_{3},X_{3},\theta \right) +\Delta \Psi ^{\dag }\left( K_{3},P_{3},X_{3},\theta \right) \right) \left( \begin{array}{c} \left( K\right) ^{-\frac{\alpha }{1+\gamma }} \\ f\left( X\right) \end{array} \right) \left( \Psi _{0}\left( K_{3},P_{3},X_{3},\theta \right) \right. \\&\left. \quad +\Delta \Psi \left( K_{3},P_{3},X_{3},\theta \right) \right) \end{aligned}$$

and is approximatively equal to:

$$\begin{aligned}&\Psi _{0}^{\dag }\left( K_{3},P_{3},X_{3},\theta \right) \left( \begin{array}{c} \left( K\right) ^{-\frac{\alpha }{1+\gamma }} \\ f\left( X\right) \end{array} \right) \Psi _{0}\left( K_{3},P_{3},X_{3},\theta \right) \\&\quad +\Delta \Psi ^{\dag }\left( K_{3},P_{3},X_{3},\theta \right) \left( \begin{array}{c} \left( K\right) ^{-\frac{\alpha }{1+\gamma }} \\ f\left( X\right) \end{array} \right) \Delta \Psi \left( K_{3},P_{3},X_{3},\theta \right) \end{aligned}$$

Given their form, \(\left( K\right) ^{-\frac{\alpha }{1+\gamma }}\) and \( f\left( X\right) \) can be considered to be close to their average \( \left\langle K\right\rangle ^{-\frac{\alpha }{1+\gamma }}\) and \(f\left( \left\langle X\right\rangle \right) \). As a consequence:

$$\begin{aligned}&\left\langle \Psi _{0}^{\dag }\left( \begin{array}{c} \left( K\right) ^{-\frac{\alpha }{1+\gamma }} \\ f\left( X\right) \end{array} \right) ,\Delta \Psi \right\rangle \simeq 0 \end{aligned}$$
(161)

for a perturbation \(\Delta \Psi \left( K,X\right) \) orthogonal to \(\Psi _{0}\left( K,X\right) \).

The second-order development of the first potential term—the constraint—in (20) for the state \(\Psi _{0}+\Delta \Psi \) is then:

$$\begin{aligned}&\int \Psi _{0}^{\dag }\left( K,X\right) \left( f^{1+\gamma }\left( X\right) -\frac{\kappa \rho ^{4}}{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\right. \nonumber \\&\quad \times \left. \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right) ^{2}\Psi _{0}\left( K,X\right) \nonumber \\&\quad +\int \Delta \Psi ^{\dag }\left( K,X\right) \left( f^{1+\gamma }\left( X\right) -\frac{\kappa \rho ^{4}}{d^{2}}\int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\right. \nonumber \\&\quad \times \left. \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right) \Delta \Psi \left( K,X\right) \nonumber \\&\quad -2\frac{\kappa \rho ^{2}}{d^{2}}\int \Psi _{0}^{\dag }\left( K,X\right) \left( \int \Psi _{0}^{\dag }\left( K_{3},X_{3}\right) \Delta \Psi ^{\dag }\left( K_{2},X_{2}\right) P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\right. \nonumber \\&\quad \times \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \Delta \Psi \left( K_{2},X_{2}\right) \Psi _{0}\left( K_{3},X_{3}\right) \nonumber \\&\quad +\left. \int \Psi _{0}^{\dag }\left( K_{2},X_{2}\right) \Delta \Psi ^{\dag }\left( K_{3},X_{3}\right) P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \right. \nonumber \\&\left. \quad \Delta \Psi \left( K_{3},X_{3}\right) \Psi _{0}\left( K_{2},X_{2}\right) \right) \nonumber \\&\quad \times \left( f^{1+\gamma }\left( X\right) -\kappa \int P_{2}\left( K_{2}\right) ^{\alpha }P_{3}\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\right. \nonumber \\&\left. \quad \left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right) \Psi _{0}\left( K,X\right) \end{aligned}$$
(162)

where as before, we define \(f\left( X\right) =PK^{-\frac{\alpha }{1+\gamma } }\). Integrals in (162) are taken over \(Z_{2}\) and \(Z_{3}\), the factor \(\hbox {d}Z_{2}\hbox {d}Z_{3}\) being implicit. To compute (162), we define several quantities. First, the average capital in state \(\Psi _{0}\):

$$\begin{aligned} \left\langle K\right\rangle _{X,0}=\int K\left| \Psi _{0}\left( K,X\right) \right| ^{2}dK \end{aligned}$$

As a consequence, in state \(\Psi _{0}\), the distribution of K is centred around \(\left\langle K\right\rangle _{X,0}\). Apart from the change of notation, \(\left\langle K\right\rangle _{X,0}\) has been computed previously in formula (140).

Similarly, in the state \(\Delta \Psi \), K is centred around:

$$\begin{aligned} \left\langle K\right\rangle _{X,1}=\int K\left| \Delta \Psi \left( K,X\right) \right| ^{2}dK \end{aligned}$$

Recall that \(\left\langle K\right\rangle _{X}\) stands for the average value of K in the full state \(\Psi _{0}\left( K,X\right) +\Delta \Psi \left( K,X\right) \). Given the orthogonality relation (161), the relation \( \left\langle K\right\rangle _{X}=\left\langle K\right\rangle _{X,0}+\left\langle K\right\rangle _{X,1}\) holds.

For both states \(\Psi _{0}\left( K,X\right) \) and \(\Delta \Psi \left( K,X\right) \), we set:

$$\begin{aligned} G_{0}\left( X_{i},X_{i}\right)= & {} \rho ^{2}\Psi _{0}^{\dag }\left( \left\langle K\right\rangle _{X_{i},0},X_{i}\right) \Psi _{0}\left( \left\langle K\right\rangle _{X_{i},0},X_{i}\right) \\ G\left( X_{i},X_{i}\right)= & {} G\left( \left( \left\langle K\right\rangle _{X_{i,1}},X_{i}\right) ,\left( \left\langle K\right\rangle _{X_{j},1},X_{i}\right) \right) \end{aligned}$$

We also define:

$$\begin{aligned} f_{0}\left( X\right)= & {} \int PK^{-\frac{\alpha }{1+\gamma }}\left| \Psi _{0}\left( K,X\right) \right| ^{2}dK\simeq \left\langle P\right\rangle _{X,0}\left\langle K^{-\frac{\alpha }{1+\gamma }}\right\rangle _{X,0}\left| \Psi _{0}\left( \left\langle K\right\rangle _{X,0},X\right) \right| ^{2} \\ f_{0}^{1+\gamma }\left( X\right)= & {} \int P^{1+\gamma }K^{-\alpha }\left| \Psi _{0}\left( K,X\right) \right| ^{2}dK\simeq \left\langle P^{1+\gamma }\right\rangle _{X,0}\left\langle K^{-\alpha }\right\rangle _{X,0}\left| \Psi _{0}\left( \left\langle K\right\rangle _{X,0},X\right) \right| ^{2} \end{aligned}$$

and:

$$\begin{aligned} f_{1}\left( X\right)= & {} \int PK^{-\frac{\alpha }{1+\gamma }}\left| \Delta \Psi \left( K,X\right) \right| ^{2}dK\simeq \left\langle P\right\rangle _{X,1}\left\langle K^{-\frac{\alpha }{1+\gamma } }\right\rangle _{X,1}\left| \Delta \Psi \left( \left\langle K\right\rangle _{X,1},X\right) \right| ^{2} \\ f_{1}^{1+\gamma }\left( X\right)= & {} \int P^{1+\gamma }K^{-\alpha }\left| \Delta \Psi \left( K,X\right) \right| ^{2}dK\simeq \left\langle P^{1+\gamma }\right\rangle _{X,1}\left\langle K^{-\alpha }\right\rangle _{X,1}\left| \Delta \Psi \left( \left\langle K\right\rangle _{X,1},X\right) \right| ^{2} \end{aligned}$$

Remark that \(f_{0}\left( X\right) \) was computed in (141).

Given these definitions, we can compute (162) in first approximation by replacing K and P by their expectations in the states \(\Psi _{0}\) and \(\Delta \Psi \), and using the Green functions. The first term is the action for the sole state \(\Psi _{0}\left( K,X\right) \) can be rewritten:

$$\begin{aligned}&\int \Psi _{0}^{\dag }\left( K,X\right) \\&\quad \left( \left( f\left( X\right) \right) ^{1+\gamma }-\frac{\kappa \rho ^{4}}{d^{2}}\int G_{0}\left( X_{2},X_{2}\right) G_{0}\left( X_{3},X_{3}\right) \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{1+\gamma }}f_{0}\left( X_{3}\right) f_{0}\left( X_{2}\right) \right. \\&\quad \times \left. \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \hbox {d}X_{2}\hbox {d}X_{3}\right) ^{2}\Psi _{0}\left( K,X\right) \end{aligned}$$

The last term in (162) rewrites in first approximation:

$$\begin{aligned}&-\frac{\kappa \rho ^{4}}{d^{2}}\int G_{0}\left( X_{2},X_{2}\right) G\left( X_{3},X_{3}\right) \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3}}^{-\frac{\alpha }{ 1+\gamma }}f_{1}\left( X_{3}\right) f_{0}\left( X_{2}\right) \nonumber \\&\quad \exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) \hbox {d}X_{2}\hbox {d}X_{3} \nonumber \\&\quad \times \left( \left( f_{0}\left( X\right) \right) ^{1+\gamma }-\frac{ \kappa \rho ^{4}}{d^{2}}\int G_{0}\left( X_{2},X_{2}\right) G_{0}\left( X_{3},X_{3}\right) \right. \nonumber \\&\quad \times \left. \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{ 1+\gamma }}f_{0}\left( X_{3}\right) f_{0}\left( X_{2}\right) \exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) G_{0}\left( X,X\right) \nonumber \\&\quad -\frac{\kappa \rho ^{4}}{d^{2}}\int G\left( X_{2},X_{2}\right) G_{0}\left( X_{3},X_{3}\right) \left\langle K\right\rangle _{X_{2}}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{ 1+\gamma }}f_{0}\left( X_{3}\right) f_{1}\left( X_{2}\right) \nonumber \\&\qquad \exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) \hbox {d}X_{2}\hbox {d}X_{3} \nonumber \\&\quad \times \left( \left( f_{0}\left( X\right) \right) ^{1+\gamma }-\frac{ \kappa \rho ^{4}}{d^{2}}\int G_{0}\left( X_{2},X_{2}\right) G_{0}\left( X_{3},X_{3}\right) \right. \nonumber \\&\quad \times \left. \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{ 1+\gamma }}f_{0}\left( X_{3}\right) f_{0}\left( X_{2}\right) \exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) G_{0}\left( X,X\right) \nonumber \\ \end{aligned}$$
(163)

We can approximate the integrals by their estimations, as we did in appendix 3. In first approximation, the exponential \(\exp \left( -\frac{\left| X_{2}-X_{3}\right| }{d}\right) \) is replaced by \(2d\delta \left( X_{2}-X_{3}\right) \) and we consider a uniform distribution for X, so that (163) is equal to:

$$\begin{aligned}&-\frac{\kappa \rho ^{4}}{2d}\int \left( \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{2}}^{-\frac{\alpha }{1+\gamma }}+\left\langle K\right\rangle _{X_{2}}^{ \frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{2},0}^{- \frac{\alpha }{1+\gamma }}\right) f_{1}\left( X_{2}\right) f_{0}\left( X_{2}\right) \hbox {d}X_{2} \nonumber \\&\qquad \times \int \exp \left( -\left( \frac{\left| X-X_{2}\right| }{d} \right) \right) \left( \left( f_{0}\left( X\right) \right) ^{1+\gamma }\right. \nonumber \\&\left. \quad - \frac{\kappa \rho ^{4}}{2d}\int \left\langle K\right\rangle _{X_{2},0}^{ \frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}\right) \right) \right) \hbox {d}X_{2} \end{aligned}$$
(164)

where we used the fact that the K-part of \(\Psi _{0}\) is peaked around \( \left\langle K\right\rangle _{X,0}\), and that the X-part is assumed to be distributed uniformly. We can approximate the exponential in (164) by its maximum value, and (164) becomes:

$$\begin{aligned}&-\frac{\kappa \rho ^{4}}{2d}\int \left( \left\langle K\right\rangle _{X,0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X}^{- \frac{\alpha }{1+\gamma }}+\left\langle K\right\rangle _{X}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X,0}^{-\frac{\alpha }{ 1+\gamma }}\right) f_{1}\left( X\right) f_{0}\left( X\right) \hbox {d}X_{2} \\&\quad \times \left( \left( f_{0}\left( X\right) \right) ^{1+\gamma }-\frac{ \kappa \rho ^{4}}{2d}\int \left\langle K\right\rangle _{X_{2},0}^{\frac{ \alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d} \right) \right) \right) \hbox {d}X_{2} \end{aligned}$$

This expression is approximately equals to 0, since the equality:

$$\begin{aligned} \left( f_{0}\left( X\right) \right) ^{1+\gamma }-\frac{\kappa \rho ^{4}}{2d} \int \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\exp \left( -\left( \frac{\left| X-X_{2}\right| }{d}\right) \right) \hbox {d}X_{2}\simeq 0\quad \end{aligned}$$

holds for X (see (138)). Thus, at the second order, the term (163) becomes:

$$\begin{aligned}&\int \Delta \Psi ^{\dag }\left( \left( f\left( X\right) \right) ^{1+\gamma }- \frac{\kappa \rho ^{4}}{2d}\int G_{0}^{2}\left( X_{2},X_{2}\right) \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\exp \left( -\frac{ \left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2}\right) ^{2} \Delta \Psi \nonumber \\ \end{aligned}$$
(165)

with, as before:

$$\begin{aligned} f^{1+\gamma }\left( X\right) =P^{1+\gamma }K^{\alpha } \end{aligned}$$

As a consequence, (162) writes:

$$\begin{aligned}&\int \Psi _{0}^{\dag }\left( K,X\right) \left( \left( f\left( X\right) \right) ^{1+\gamma }-\frac{\kappa \rho ^{4}}{2d}\int G_{0}^{2}\left( X_{2},X_{2}\right) \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\right. \nonumber \\&\left. \quad \exp \left( -\frac{\left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2}\right) ^{2}\Psi _{0}\left( K,X\right) \nonumber \\&\qquad +\int \Delta \Psi ^{\dag }\left( \left( f\left( X\right) \right) ^{1+\gamma }-\frac{\kappa \rho ^{4}}{2d}\int G_{0}^{2}\left( X_{2},X_{2}\right) \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\right. \nonumber \\&\left. \quad \times \exp \left( -\frac{\left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2}\right) ^{2}\Delta \Psi \nonumber \\&\quad =\int \Psi ^{\dag }\left( K,X\right) \left( \left( f\left( X\right) \right) ^{1+\gamma }-\frac{\kappa \rho ^{4}}{2d}\int G_{0}^{2}\left( X_{2},X_{2}\right) \right. \nonumber \\&\qquad \times \left. \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma } }\left( f_{0}\left( X_{2}\right) \right) ^{2}\exp \left( -\frac{\left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2}\right) ^{2}\Psi \left( K,X\right) \end{aligned}$$
(166)

where (161) has been used. The defining equation for \(f\left( X\right) \) is thus:

$$\begin{aligned} \left( f\left( X\right) \right) ^{1+\gamma }=\frac{\kappa \rho ^{4}}{2d}\int G_{0}^{2}\left( X_{2},X_{2}\right) \left\langle K\right\rangle _{X_{2},0}^{ \frac{\alpha \left( \gamma -1\right) }{1+\gamma }}\left( f_{0}\left( X_{2}\right) \right) ^{2}\exp \left( -\frac{\left| X-X_{2}\right| }{d }\right) \hbox {d}X_{2} \end{aligned}$$
(167)

which is (138), and where the assumption of a uniform distribution for X has been used.

The second potential term in (20) is written:

$$\begin{aligned}&\int \Psi ^{\dag }\left( K,X\right) \nonumber \\&\quad \times \left( \delta K-APK^{\alpha }\left( 1- \frac{\kappa \rho ^{4}}{d^{2}}\int \frac{P_{3}\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{1}-X_{3}\right| }{d}\right) }{ P_{2}^{\gamma }}\left| \Psi \left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi \left( K_{3},X_{3}\right) \right| ^{2}\right) \right) ^{2} \Psi \left( K,X\right) \nonumber \\ \end{aligned}$$
(168)

Here again, the integration factor \(\hbox {d}Z_{2}\hbox {d}Z_{3}\) is omitted. At the second order in \(\left( \Delta \Psi ,\Delta \Psi ^{\dag }\right) \), (168) is given by:

$$\begin{aligned}&\int \Psi _{0}^{\dag }\left( K,X\right) \left( \delta K-APK^{\alpha }\right. \nonumber \\&\quad \times \left. \left( 1-\frac{\kappa \rho ^{4}}{d^{2}}\int \frac{P_{3}\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) }{P_{2}^{\gamma }}\left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right) \right) ^{2} \Psi _{0}\left( K,X\right) \nonumber \\&\quad +\int \Delta \Psi ^{\dag }\left( K,X\right) \left( \delta K-APK^{\alpha }\right. \nonumber \\&\quad \times \left. \left( 1-\frac{\kappa \rho ^{4}}{d^{2}}\int \frac{P_{3}\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) }{P_{2}^{\gamma }}\left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right) \right) ^{2} \Delta \Psi \left( K,X\right) \nonumber \\&\quad +\int \Psi _{0}^{\dag }\left( K,X\right) \left( \int \frac{\exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) }{P_{2}^{\gamma }}P_{3}\left| \Delta \Psi \left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right. \nonumber \\&\quad \left. +\int \frac{\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) }{P_{2}^{\gamma }} P_{3}\left| \Delta \Psi \left( K_{3},X_{3}\right) \right| ^{2}\left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\right) \nonumber \\&\quad \times \left( \delta K-APK^{\alpha }\left( 1-\frac{\kappa \rho ^{4}}{d^{2}} \int \frac{P_{3}\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) }{P_{2}^{\gamma }}\left| \Psi _{0}\left( K_{2},X_{2}\right) \right| ^{2}\left| \Psi _{0}\left( K_{3},X_{3}\right) \right| ^{2}\right) \right) \Psi _{0}\left( K,X\right) \nonumber \\ \end{aligned}$$
(169)

The first term computes the potential in the state \(\Psi _{0}\). Under our assumptions, the last term of (169) rewrites:

$$\begin{aligned}&\int \left( \int G_{0}\left( X_{3},X_{3}\right) G\left( X_{2},X_{2}\right) \left( \left\langle K\right\rangle _{X_{3},0}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( \left\langle K\right\rangle _{X_{2},1}\right) ^{-\frac{ \alpha }{1+\gamma }}f_{0}\left( X_{3}\right) \left( f_{1}\left( X_{2}\right) \right) ^{-\gamma }\right. \\&\quad \times \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \hbox {d}X_{2}\hbox {d}X_{3} \\&\quad +\int G_{0}\left( X_{2},X_{2}\right) G\left( X_{3},X_{3}\right) \left( \left\langle K\right\rangle _{X_{3},1}\right) ^{\frac{\alpha \gamma }{ 1+\gamma }}\left( \left\langle K\right\rangle _{X_{2},0}\right) ^{-\frac{ \alpha }{1+\gamma }} \\&\quad \left. \times f_{1}\left( X_{3}\right) \left( f_{0}\left( X_{2}\right) \right) ^{-\gamma }\exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \hbox {d}X_{2}\hbox {d}X_{3}\right) \\&\quad \times \left( \delta \left\langle K\right\rangle _{X,0}-AP\left\langle K\right\rangle _{X,0}^{\alpha }\right. \\&\quad \times \left. \left( 1-\frac{\kappa \rho ^{4}}{4d^{2}}\int \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{1+\gamma }}f_{0}\left( X_{3}\right) \left( f_{0}\left( X_{2}\right) \right) ^{-\gamma }\right. \right. \\&\left. \left. \quad \exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) G_{0}\left( X,X\right) \end{aligned}$$

and is null in first approximation, since:

$$\begin{aligned} 0= & {} \delta \left\langle K\right\rangle _{X,0}-AP\left\langle K\right\rangle _{X,0}^{\alpha } \nonumber \\&\times \left( 1-\frac{\kappa \rho ^{4}}{4d^{2}}\int \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma } }\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\right. \nonumber \\&\left. \exp \left( - \frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d} \right) \hbox {d}X_{2}\hbox {d}X_{3}\right) \end{aligned}$$
(170)

holds for all X [see (102), applied in the state \(\Psi _{0}\)]. Thus, the second-order term of the potential in (168) becomes:

$$\begin{aligned}&\int \Delta \Psi ^{\dag }\left( K,X\right) \left( \delta K-APK^{\alpha }\left( 1-\frac{\kappa \rho ^{4}}{4d^{2}}\int \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\right. \right. \nonumber \\&\quad \times \left. \left. \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \hbox {d}X_{2}\right) \right) ^{2}\Delta \Psi \left( K,X\right) \end{aligned}$$
(171)

Moreover, as a consequence of (170), the potential (168) writes:

$$\begin{aligned}&\int \left( \Psi _{0}+\Delta \Psi \right) ^{\dag }\left( \delta K-APK^{\alpha }\left( 1-\frac{\kappa \rho ^{4}}{4d^{2}}\int \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X_{3},0}^{-\frac{\alpha }{1+\gamma }}f\left( X_{3}\right) \left( f\left( X_{2}\right) \right) ^{-\gamma }\right. \right. \nonumber \\&\qquad \times \left. \left. \exp \left( -\frac{\left| X-X_{2}\right| +\left| X_{2}-X_{3}\right| }{d}\right) \hbox {d}X_{2}\hbox {d}X_{3}\right) \right) ^{2}\left( \Psi _{0}+\Delta \Psi \right) \nonumber \\&\quad \simeq \int \left( \Psi _{0}+\Delta \Psi \right) ^{\dag }\left( \delta K-APK^{\alpha }\left( 1-\frac{\kappa \rho ^{4}}{2d}\int \left\langle K\right\rangle _{X_{2},0}^{\frac{\alpha \left( \gamma -1\right) }{1+\gamma } }\left( f\left( X_{2}\right) \right) ^{1-\gamma }\right. \right. \nonumber \\&\quad \left. \left. \times \exp \left( -\frac{ \left| X-X_{2}\right| }{d}\right) \hbox {d}X_{2}\right) \right) ^{2}\left( \Psi _{0}+\Delta \Psi \right) \nonumber \\&\quad =\int \left( \Psi _{0}+\Delta \Psi \right) ^{\dag }\delta ^{2}\left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}\left( \Psi _{0}+\Delta \Psi \right) \end{aligned}$$
(172)

Equations (172) and (167) mean that \(f\left( X\right) \simeq f_{0}\left( X\right) \) and \(\left\langle K\right\rangle _{X}\simeq \left\langle K\right\rangle _{X,0}\). These two quantities have been computed previously in this appendix. The corrections due to \(\Delta \Psi \left( K,X\right) \) can be neglected in first approximation. We will use this quadratic approximation for the potential to compute the Green function in phase 2.

Ultimately, recall that the potentials (166) and (172) have been computed for \(\Delta \Psi \) orthogonal to \(\Psi _{0}\). To compute the second-order potential, we have to introduce a fluctuation proportional to \( \Psi _{0}\), namely \(\Delta \Psi =\frac{\left( \delta \rho \right) ^{2}}{ 2\rho ^{2}}\Psi _{0}\) corresponding to a variation \(\rho ^{2}\rightarrow \rho ^{2}+\left( \delta \rho \right) ^{2}\). The associated variation of \( S\left( \Psi \right) \) is \(\frac{1}{2}\frac{\partial S\left( \Psi _{0}\right) }{\partial \rho ^{2}}\left( \delta \rho \right) ^{2}\). A sufficient first approximation can be found using (95) in appendix 2. This equation states that:

$$\begin{aligned} S\left( \Psi \right) \simeq \left( \frac{1}{2}\kappa _{0}^{\frac{1}{2} }-\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}\right) \rho ^{2} \end{aligned}$$

for \(\Psi \) proportional to \(\Psi _{0}\). Thus:

$$\begin{aligned} \frac{\partial S\left( \Psi _{0}\right) }{\partial \rho ^{2}}\left( \delta \rho \right) ^{2}=\left( 2\kappa _{2}\rho ^{2}-\kappa _{1}\right) \rho ^{2}\left( \delta \rho \right) ^{2} \end{aligned}$$
(173)

since \(\frac{1}{2}\kappa _{0}^{\frac{1}{2}}-\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}=0\) for \(\Psi _{0}\). Writing:

$$\begin{aligned} \Delta \Psi =\Delta ^{\prime }\Psi +\frac{\left( \delta \rho \right) ^{2}}{ 2\rho ^{2}}\Psi _{0} \end{aligned}$$
(174)

where \(\Delta ^{\prime }\Psi \) is orthogonal to \(\Psi _{0}\), we can also write (173) as:

$$\begin{aligned} \frac{1}{2}\frac{\partial S\left( \Psi _{0}\right) }{\partial \rho ^{2}} \left( \delta \rho \right) ^{2}=\left( 2\kappa _{2}\rho ^{2}-\kappa _{1}\right) \rho ^{2}\left| \int \Delta \Psi \left( K,X,\theta \right) \Psi _{0}\left( K,X,\theta \right) \right| ^{2} \end{aligned}$$
(175)

Formulas (172) and (175) will be used to compute the quadratic action for phase 2.

Corrections of order \(\frac{A^{2}}{\bar{A}^{2}}\) to the potential

We can include, as we did in the first phase, the first-order corrections in \(\frac{A^{2}}{\bar{A}^{2}}\). Since the terms appearing in the second-order expansion are (165) and (171), the potential becomes:

$$\begin{aligned} \int \delta ^{2}\left( 1+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}\right) \Delta \Psi ^{\dag }\left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}\Delta \Psi \end{aligned}$$
(176)

Appendix 5

In this section, we compute the effective quadratic action and the Green functions in both phases. Recall that, deriving the average capital level \( \left\langle K\right\rangle _{X}\) and the price level P, we obtained the quadratic approximation of the potential term for the capital. In phase 1, we had (135):

$$\begin{aligned} \delta ^{2}\left( 1+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}\right) \left( K-\left\langle K\right\rangle _{X}\right) ^{2} \end{aligned}$$
(177)

In phase 2, (176) holds:

$$\begin{aligned} \int \delta ^{2}\left( 1+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}\right) \Delta \Psi ^{\dag }\left( \delta K-\left\langle K\right\rangle _{X,0}\right) ^{2}\Delta \Psi \end{aligned}$$
(178)

where U stands for \(\left\langle U\right\rangle \) defined in (128). For each phase, we first compute the quadratic effective action and then derive the Green function.

Case \(\rho =0\)

Effective quadratic action For \(\rho =0\), using (177) and discarding the X contribution, the action is:

$$\begin{aligned}&\int \Psi ^{\dag }\left( K,P,X,\theta \right) \left( -\frac{\sigma ^{2}}{2} \nabla _{K}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}+\frac{1}{ 2\sigma ^{2}}\left( \delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+ \bar{A}^{2}\right) ^{2}}\right) \right. \nonumber \\&\left. \quad \left( K-\left\langle K\right\rangle \right) ^{2}+\frac{1}{2\vartheta ^{2}}+\alpha \right) \Psi \left( K,P,X,\theta \right) \end{aligned}$$
(179)

whose Green function is given by:

$$\begin{aligned}&G_{0}\left( K,K^{\prime },\theta ,\theta ^{\prime },t\right) \\&\quad =\sqrt{\frac{ \omega }{2\pi \sigma ^{2}\sinh \left( \omega t\right) }}\exp \left( \left( - \frac{\omega }{2\sigma ^{2}\sinh \left( \omega t\right) }\right) \left( \left( K^{2}+\left( K^{\prime }\right) ^{2}\right) \right. \right. \left. \left. \cosh \left( \omega t\right) -2KK^{\prime }\right) \right) \\&\qquad \times \sqrt{\frac{1}{2\pi \vartheta ^{2}t}}\exp \left( -\frac{\left( \theta ^{\prime }-\theta \right) ^{2}}{2\vartheta ^{2}t}\right) \end{aligned}$$

with:

$$\begin{aligned} \omega ^{2}=\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}} \end{aligned}$$

To include the X-dependent part of the action (in the sequel, the dependency of \(\Psi \) in \(\theta \) is understood):

$$\begin{aligned}&\int \Psi ^{\dag }\left( K,X\right) \left( -\frac{\sigma _{X}^{2}}{2} \nabla _{X}^{2}+\frac{\kappa _{0}}{2\sigma _{X}^{2}}X^{2}\right) \Psi \left( K,X\right) \nonumber \\&\quad -\int \frac{\kappa _{1}}{2}\left( \Psi ^{\dag }\left( K,X\right) \Psi \left( K,X\right) \right) \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left( \Psi ^{\dag }\left( K^{\prime },Y\right) \Psi \left( K^{\prime },Y\right) \right) \nonumber \\&\quad +\int \frac{\kappa _{2}}{3}\left( \Psi ^{\dag }\left( K,X\right) \Psi \left( K,X\right) \right) \nonumber \\&\qquad \times \exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K'',Z\right) \right| ^{2} \nonumber \\ \end{aligned}$$
(180)

and to account for its contribution to the Green function, we replace the interaction potential by its average in variables Y and Z. As a consequence, in (180):

$$\begin{aligned} -\frac{\kappa _{1}}{2}\left| \Psi \left( K,X\right) \right| ^{2} \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{ \left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2} \end{aligned}$$

and:

$$\begin{aligned}&\frac{\kappa _{2}}{3}\left| \Psi \left( K,X\right) \right| ^{2}\exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \\&\quad \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K'',Z\right) \right| ^{2} \end{aligned}$$

are replaced by:

$$\begin{aligned}&-\frac{\kappa _{1}}{2}\left\langle \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\right\rangle \left| \Psi \left( K,X\right) \right| ^{2} \end{aligned}$$
(181)

and:

$$\begin{aligned}&\frac{\kappa _{2}}{3}\left| \Psi \left( K,X\right) \right| ^{2}\nonumber \\&\quad \left\langle \exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K'',Z\right) \right| ^{2}\right\rangle ,\nonumber \\ \end{aligned}$$
(182)

respectively.

To compute (181), we use the fact that, at the lowest order in perturbation:

$$\begin{aligned}&\left\langle \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\right\rangle \\&\quad \simeq \int \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}G\left( \left( K^{\prime },Y\right) ,\left( K^{\prime },Y\right) \right) dK^{\prime }dY \\&\quad \simeq \int \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}G_{Y}\left( K^{\prime },K^{\prime }\right) G\left( Y,Y\right) dK^{\prime }dY \end{aligned}$$

The hypotheses are the same as in appendix 3, that is, \(K^{\prime }\) is spread around \(\left\langle K\right\rangle _{Y}\) and the X-part of the Green function is approximatively uniformly distributed on the interval \( \left[ -1,1\right] \). The previous expression thus becomes:

$$\begin{aligned} \left\langle \int \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2}dK^{\prime }dY\right\rangle \simeq \frac{K}{2\left\langle K\right\rangle _{X}}\int \exp \left( -\chi _{1}\left| X-Y\right| \right) dY \end{aligned}$$
(183)

This last integral is computed for \(\chi _{1}<<1\) as:

$$\begin{aligned}&\int _{-1}^{1}\exp \left( -\left( \chi _{1}\right) \left| X-Y\right| \right) dY \\&\quad =\int _{-1-X}^{1-X}\exp \left( -\chi _{1}\left| u\right| \right) du=\int _{-1-X}^{0}\frac{\exp \left( \chi _{1}u\right) }{\sqrt{\alpha }} du+\int _{0}^{1-X}\frac{\exp \left( -\chi _{1}u\right) }{\sqrt{\alpha }}du \\&\quad =\frac{\left( 2-\exp \left( -\chi _{1}\left( 1+X\right) \right) -\exp \left( -\chi _{1}\left( 1-X\right) \right) \right) }{\chi _{1}} \\&\quad \simeq \frac{2\left( 1-\exp \left( -\chi _{1}\right) \right) }{\chi _{1}} -\exp \left( -\chi _{1}\right) \chi _{1}X^{2} \\&\quad \simeq 2-\chi _{1}+\frac{1}{3}\chi _{1}^{2}+\left( \chi _{1}^{2}-\chi _{1}\right) X^{2} \end{aligned}$$

and (183) becomes:

$$\begin{aligned}&\left\langle \int \frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle _{X}\left\langle K\right\rangle _{Y}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2}dK^{\prime }dY\right\rangle \nonumber \\&\quad \simeq \frac{K}{2\left\langle K\right\rangle _{X}}\left( \frac{2\left( 1-\exp \left( -\chi _{1}\right) \right) }{\chi _{1} }-\exp \left( -\chi _{1}\right) \chi _{1}X^{2}\right) \end{aligned}$$
(184)

By the same token, we compute the contribution (182) by writing:

$$\begin{aligned}&\int \kappa _{2}\left\langle \exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K^{\prime \prime },Z\right) \right| ^{2}\right\rangle \\&\quad dK^{\prime }dK^{\prime \prime }\hbox {d}Y\hbox {d}Z \\&\quad \simeq \int \kappa _{2}\exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) G\left( Y,Y\right) G\left( Z,Z\right) \hbox {d}Y\hbox {d}Z \\&\quad \simeq \int \frac{\kappa _{2}}{4}\exp \left( -\chi _{2}\left( \left| X-Y\right| +\left| X-Z\right| +\left| Y-Z\right| \right) \right) \hbox {d}Y\hbox {d}Z \end{aligned}$$

where we used that:

$$\begin{aligned} G\left( \left( K^{\prime },Y\right) ,\left( K^{\prime },Y\right) \right) =G_{Y}\left( K^{\prime },K^{\prime }\right) G\left( Y,Y\right) \end{aligned}$$

and \(\int G_{Y}\left( K^{\prime },K^{\prime }\right) dK^{\prime }\) has been normalised to 1.

As a consequence, replacing \(\exp \left( -\chi _{2}\left| Y-Z\right| \right) \) by \(\frac{2\left( 1-\exp \left( -\chi _{2}\right) \right) \delta \left( Y-Z\right) }{\chi _{2}}\) yields:

$$\begin{aligned}&\int \kappa _{2}\left\langle \exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K^{\prime \prime },Z\right) \right| ^{2}\right\rangle \nonumber \\&\qquad dK^{\prime }dK^{\prime \prime }\hbox {d}Y\hbox {d}Z \nonumber \\&\quad \simeq \int \frac{\kappa _{2}}{2}\frac{\left( 1-\exp \left( -\chi _{2}\right) \right) }{\chi _{2}}\exp \left( -2\chi _{2}\left( \left| X-Y\right| \right) \right) dY \nonumber \\&\quad \simeq \frac{\kappa _{2}\left( 1-\exp \left( -\chi _{2}\right) \right) }{ 2\chi _{2}}\left( \frac{\left( 1-\exp \left( -2\chi _{2}\right) \right) }{ \chi _{2}}-2\exp \left( -2\chi _{2}\right) \chi _{2}X^{2}\right) \nonumber \\&\quad \simeq \kappa _{2}\left( 1-\frac{3}{2}\chi _{2}+\frac{4}{3}\kappa _{2}\chi _{2}^{2}\right) +\kappa _{2}\left( \frac{5}{2}\chi _{2}^{2}-\chi _{2}\right) X^{2} \end{aligned}$$
(185)

Under the approximations (183) and (185), the X-dependent part of the action becomes:

$$\begin{aligned} \int \left( \Psi ^{\dag }\left( X\right) \left( -\frac{\sigma _{X}^{2}}{2} \nabla _{X}^{2}\right) \Psi \left( X\right) +\frac{\omega _{X}^{2}}{2\sigma _{X}^{2}}\Psi ^{\dag }\left( X\right) X^{2}\Psi \left( X\right) +\alpha _{X}\Psi ^{\dag }\left( X\right) \Psi \left( X\right) \right) \end{aligned}$$
(186)

where:

$$\begin{aligned} \omega _{X}^{2}=\kappa _{0}+\left( \frac{\kappa _{1}}{2}\frac{K}{ \left\langle K\right\rangle _{X}}\left( \chi _{1}-\chi _{1}^{2}\right) - \frac{\kappa _{2}}{3}\left( 2\chi _{2}-5\chi _{2}^{2}\right) \right) \sigma _{X}^{2} \end{aligned}$$
(187)

and:

$$\begin{aligned} \alpha _{X}=\alpha +\frac{1}{2\vartheta ^{2}}-\frac{\kappa _{1}}{2}\frac{K}{ \left\langle K\right\rangle _{X}}\left( 1-\frac{\chi _{1}}{2}+\frac{1}{6} \chi _{1}^{2}\right) +\frac{\kappa _{2}}{3}\left( 1-\frac{3}{2} \chi _{2}+\frac{4}{3}\chi _{2}^{2}\right) \end{aligned}$$
(188)

As a consequence, gathering (179) and (186), the overall second-order action becomes:

$$\begin{aligned}&\int \Psi ^{\dag }\left( K,X,\theta \right) \left( -\frac{\sigma ^{2}\nabla _{K}^{2}}{2}-\frac{\vartheta ^{2}\nabla _{\theta }^{2}}{2}-\frac{ \sigma _{X}^{2}\nabla _{X}^{2}}{2}\right. \nonumber \\&\quad \left. +\frac{1}{2}\left( \delta ^{2}+\frac{\bar{A}^{2}A^{2} }{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}\right) \left( K-\left\langle K\right\rangle \right) ^{2}+\frac{\omega _{X}^{2}X^{2}}{2\sigma _{X}^{2}} +\alpha _{X}\right) \Psi \left( K,X,\theta \right) \nonumber \\ \end{aligned}$$
(189)

This quadratic action will be used to compute the Green functions. To do so, we must include an exponential factor induced by the change of variable (82).

Exponential factor The change of variable (82) modifies the Green functions and includes a factor:

$$\begin{aligned} \exp \left( -\int \left( \frac{\left( \delta K-APK^{\alpha }+U_{1}\right) }{ \sigma ^{2}}\right) \right) \Psi \left( K,P,X,\theta \right) \end{aligned}$$

with:

$$\begin{aligned}&U_{1}=\frac{\kappa }{d^{2}}APK^{\alpha }\int \frac{P_{3}\exp \left( -\left( \left| X-X_{2}\right| +\left| X-X_{3}\right| \right) \right) }{P_{2}^{\gamma }}\left| \Psi \left( K_{2},P_{2},X_{2},\theta \right) \right| ^{2}\\&\quad \times \left| \Psi \left( K_{3},P_{3},X_{3},\theta \right) \right| ^{2}\hbox {d}Z_{2}\hbox {d}Z_{3} \end{aligned}$$

Given that:

$$\begin{aligned}&\int \left( APK^{\alpha }-U_{1}\right) \\&\quad \simeq \int APK^{\alpha }\left( 1-\frac{\kappa }{d^{2}}\left\langle \int \frac{P_{3}\exp \left( -\left( \left| X_{1}-X_{2}\right| +\left| X_{1}-X_{3}\right| \right) \right) }{P_{2}^{\gamma }}\left| \Psi \left( K_{2},P_{2},X_{2},\theta \right) \right| ^{2}\right. \right. \\&\left. \left. \qquad \left| \Psi \left( K_{3},P_{3},X_{3},\theta \right) \right| ^{2}\right\rangle \right) \\&\quad =\int \kappa APK^{\alpha }U \end{aligned}$$

the exponential factor rewrites:

$$\begin{aligned}&\exp \left( -\int \left( \frac{\left( \delta K-APK^{\alpha }+V_{1}\right) }{ \sigma ^{2}}\right) \right) \\&\quad =\exp \left( -\int \left( \frac{\left( \delta K-APK^{\alpha }U\right) }{\sigma ^{2}}\right) \right) \\&\quad = \exp \left( -\int \left( \frac{\left( \delta K-A\left( K\right) ^{\frac{ \alpha \gamma }{1+\gamma }}f\left( X\right) U\right) }{\sigma ^{2}}\right) \right) \end{aligned}$$

We have seen in appendix 3 that:

$$\begin{aligned} \delta \left\langle K\right\rangle _{X}-A\left( \left\langle K\right\rangle _{X}\right) ^{\frac{\alpha \gamma }{1+\gamma }}f\left( X\right) U=0 \end{aligned}$$

so that we can rewrite the term in the exponential as:

$$\begin{aligned} \frac{\delta }{\sigma ^{2}}\int \left( K-\left( \frac{K}{\left\langle K\right\rangle _{X}}\right) ^{\frac{\alpha \gamma }{1+\gamma }}\left\langle K\right\rangle _{X}\right)\simeq & {} \frac{\delta }{\sigma ^{2}}\int \left( K-\left\langle K\right\rangle _{X}\right) \nonumber \\= & {} \frac{\delta \left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{ 2\sigma ^{2}} \end{aligned}$$
(190)

for \(\gamma<<1\).

Computation of the Green function The action (189) is now quadratic, but the variables K and X are entangled via \(\omega \) and \(\omega _{X}\). To find the Green function between (KX) and \((K^{\prime },X^{\prime })\), we simplify the problem by replacing (KX) in \(\omega \) and \(\omega _{X}\) by their average trajectory values. In first approximation, it amounts to replacing any expression f(KX) by its average \(\frac{f(K,X)+f(K^{\prime },X^{\prime })}{ 2}\). We then set in (189):

$$\begin{aligned} \left\langle K\right\rangle= & {} \frac{\left\langle K\right\rangle _{X}+\left\langle K\right\rangle _{X^{\prime }}}{2} \end{aligned}$$
(191)
$$\begin{aligned} \bar{\omega }_{X}^{2}= & {} \kappa _{0}+\left( \frac{\kappa _{1}}{4}\left( \frac{K}{ \left\langle K\right\rangle _{X}}+\frac{K^{\prime }}{\left\langle K\right\rangle _{X^{\prime }}}\right) \left( \chi _{1}-\chi _{1}^{2}\right) - \frac{\kappa _{2}}{3}\left( 2\chi _{2}-5\chi _{2}^{2}\right) \right) \sigma _{X}^{2} \end{aligned}$$
(192)
$$\begin{aligned} \bar{\alpha }_{X}= & {} \alpha +\frac{1}{2\vartheta ^{2}}-\frac{\kappa _{1}}{4} \left( \frac{K}{\left\langle K\right\rangle _{X}}+\frac{K^{\prime }}{ \left\langle K\right\rangle _{X^{\prime }}}\right) \left( 1-\frac{\chi _{1}}{ 2}+\frac{1}{6}\chi _{1}^{2}\right) +\frac{\kappa _{2}}{3}\left( 1-\frac{3}{2}\chi _{2}+\frac{4}{3}\chi _{2}^{2}\right) \end{aligned}$$
(193)
$$\begin{aligned} \bar{U}= & {} 1-h\exp \left( -\frac{\left| X\right| +\left| X^{\prime }\right| }{2d}\right) \left( 1-\frac{\cosh \frac{X}{d}+\cosh \frac{X^{\prime }}{d}}{2\exp \left( \frac{1}{d}\right) }\right) \nonumber \\ \bar{\omega }= & {} \sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}\bar{U} ^{2}+\bar{A}^{2}\right) ^{2}}} \end{aligned}$$
(194)

The value of h is given by (143). If we replace \(\left| X\right| \) and \(\left| X^{\prime }\right| \) by their average value \(\frac{1}{2}\), we have:

$$\begin{aligned} \bar{U}&=U=1-h\exp \left( -\frac{1}{d}\right) \left( 1-\frac{\cosh \frac{1}{ 2d}}{\exp \left( \frac{1}{d}\right) }\right) \nonumber \\ \bar{\omega }&\simeq \omega =\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{ \left( A^{2}\bar{U}^{2}+\bar{A}^{2}\right) ^{2}}} \end{aligned}$$
(195)

Using these average values, and including the exponential factor (190), the Green function of the action (189) can be computed. It is the Laplace transform of the following temporal transition function, with parameter \(\bar{\alpha }_{X}\simeq \alpha \) and small coupling parameters:

$$\begin{aligned}&G\left( K,K^{\prime },P,P^{\prime },X,X^{\prime },\theta ,\theta ^{\prime },t\right) \\&\quad =\exp \left( -\left[ \frac{\delta \left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{2\sigma ^{2}}\right] _{\left( K,X\right) }^{\left( K^{\prime },X^{\prime }\right) }\right) \\&\qquad \times \sqrt{\frac{\bar{\omega }/2\pi \sigma ^{2}}{\sinh \left( \bar{\omega } t\right) }}\exp \left( -\frac{\bar{\omega }\left( \left( \left( K-\left\langle K\right\rangle \right) ^{2}+\left( K^{\prime }-\left\langle K\right\rangle \right) ^{2}\right) \cosh \left( \bar{\omega }t\right) -2\left( K-\left\langle K\right\rangle \right) \left( K^{\prime }-\left\langle K\right\rangle \right) \right) }{2\sigma ^{2}\sinh \left( \bar{\omega }t\right) }\right) \\&\qquad \times \sqrt{\frac{\bar{\omega }_{X}/2\pi \sigma _{X}^{2}}{\sinh \left( \bar{\omega }_{X}t\right) }}\exp \left( -\frac{\bar{\omega }_{X}\left( \left( X^{2}+\left( X^{\prime }\right) ^{2}\right) \cosh \left( \bar{\omega } _{X}t\right) -2XX^{\prime }\right) }{2\sigma _{X}^{2}\sinh \left( \bar{\omega }_{X}t\right) }\right) \\&\qquad \times \sqrt{\frac{1}{2\pi \vartheta ^{2}t}}\exp \left( -\frac{\left( \theta ^{\prime }-\theta -t\right) ^{2}}{2\vartheta ^{2}t}\right) \times \delta \left( P-\frac{D\exp \left( -\frac{\left| X\right| }{1+\gamma }\right) }{\left( K\right) ^{\frac{\alpha }{1+\gamma }}}\right) \delta \left( P^{\prime }-\frac{D\exp \left( -\frac{\left| X^{\prime }\right| }{1+\gamma }\right) }{\left( K^{\prime }\right) ^{\frac{\alpha }{1+\gamma }}}\right) \end{aligned}$$

For \(\vartheta ^{2}<<1\), the variable t can be replaced by \(\theta ^{\prime }-\theta \), for \(\theta ^{\prime }>\theta \). Actually, due to the term:

$$\begin{aligned} \exp \left( -\frac{\left( \theta ^{\prime }-\theta -t\right) ^{2}}{ 2\vartheta ^{2}t}\right) \end{aligned}$$
(196)

the Green function is non-null for values of \(\theta \) and \(\theta ^{\prime } \) such that \(\theta ^{\prime }-\theta -t=0\). Since \(t>0\), this implies that the replacement is only valid for \(\theta ^{\prime }>\theta \); otherwise, the Green function is equal to 0. As a consequence, we can remove the time dependency in the Green function to obtain:

$$\begin{aligned}&G\left( K,K^{\prime },P,P^{\prime },X,X^{\prime },\theta ,\theta ^{\prime }\right) =\exp \left( -\left[ \frac{\delta \left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{2\sigma ^{2}}\right] _{\left( K,X\right) }^{\left( K^{\prime },X^{\prime }\right) }\right) \\&\qquad \times \sqrt{\frac{\bar{\omega }}{2\pi \sigma ^{2}\sinh \left( \bar{\omega } \left( \theta ^{\prime }-\theta \right) \right) }}\\&\qquad \exp \left( -\frac{\bar{ \omega }\left( \left( \left( K-\left\langle K\right\rangle \right) ^{2}+\left( K^{\prime }-\left\langle K\right\rangle \right) ^{2}\right) \cosh \left( \bar{\omega }\left( \theta ^{\prime }-\theta \right) \right) -2\left( K-\left\langle K\right\rangle \right) \left( K^{\prime }-\left\langle K\right\rangle \right) \right) }{2\sigma ^{2}\sinh \left( \bar{\omega }\left( \theta ^{\prime }-\theta \right) \right) }\right) \\&\qquad \times \sqrt{\frac{\bar{\omega }_{X}}{2\pi \sigma _{X}^{2}\sinh \left( \bar{ \omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) }}\exp \left( - \frac{\bar{\omega }_{X}\left( \left( X^{2}+\left( X^{\prime }\right) ^{2}\right) \cosh \left( \bar{\omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) -2XX^{\prime }\right) }{2\sigma _{X}^{2}\sinh \left( \bar{ \omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) }\right) \\&\qquad \times \delta \left( P-\frac{D\exp \left( -\frac{\left| X\right| }{ 1+\gamma }\right) }{\left( K\right) ^{\frac{\alpha }{1+\gamma }}}\right) \delta \left( P^{\prime }-\frac{D\exp \left( -\frac{\left| X^{\prime }\right| }{1+\gamma }\right) }{\left( K^{\prime }\right) ^{\frac{\alpha }{1+\gamma }}}\right) H\left( \theta ^{\prime }-\theta \right) \end{aligned}$$

where \(H\left( \theta ^{\prime }-\theta \right) \) is the Heaviside function. Replacing \(\bar{\omega }\) by \(\omega \) (see (195)) yields the formula of the text.

Note that in (196), for \(\theta ^{\prime }-\theta \rightarrow 0\), the dominant part becomes \(\frac{t}{2\vartheta ^{2}}\). This means that t can replaced in average by \(\vartheta ^{2}\), which yields the description in terms of harmonic oscillators used in appendix 3.

Case \(\rho \ne 0\)

Effective quadratic action For \(\rho \ne 0\), the K-part of the second-order expansion of the action around \(\Psi _{0}\) is obtained by using (178):

$$\begin{aligned}&\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( -\frac{ \sigma ^{2}}{2}\nabla _{K}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}- \frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}\right) \Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\&\quad +\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( \frac{ 1}{2\sigma ^{2}}\left( \delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+ \bar{A}^{2}\right) ^{2}}\right) \left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}+\alpha +\frac{1}{2\vartheta ^{2}}\right) \nonumber \\&\qquad \Delta ^{\prime }\Psi \left( K,X,\theta \right) \end{aligned}$$
(197)

where \(\Delta ^{\prime }\Psi \) is a variation orthogonal to \(\Psi _{0}\), and \(\Delta \Psi =\Delta ^{\prime }\Psi +\frac{\left( \delta \rho \right) ^{2}}{2\rho ^{2}}\Psi _{0}\) (see 174).

As explained in appendix 4, the averages are computed in state \(\Psi _{0}\). We deal with the X-part of the action as in the previous case.

$$\begin{aligned}&\int \Psi ^{\dag }\left( K,X\right) \left( -\frac{\sigma _{X}^{2}}{2} \nabla _{X}^{2}+\frac{\kappa _{0}}{2\sigma _{X}^{2}}X^{2}\right) \Psi \left( K,X\right) \nonumber \\&\quad -\frac{\kappa _{1}}{2}\int \left| \Psi \left( K,X\right) \right| ^{2}\frac{KK^{\prime }\exp \left( -\chi _{1}\left| X-Y\right| \right) }{\left\langle K\right\rangle ^{2}}\left| \Psi \left( K^{\prime },Y\right) \right| ^{2} \nonumber \\&\quad +\int \frac{\kappa _{2}}{3}\exp \left( -\chi _{2}\left| X-Y\right| -\chi _{2}\left| X-Z\right| -\chi _{2}\left| Y-Z\right| \right) \left| \Psi \left( K,X\right) \right| ^{2}\nonumber \\&\qquad \left| \Psi \left( K^{\prime },Y\right) \right| ^{2}\left| \Psi \left( K^{''},Z\right) \right| ^{2} \end{aligned}$$
(198)

We obtain the second-order expansion:

$$\begin{aligned} \int \Delta ^{\prime }\Psi ^{\dag } \left( K,X,\theta \right) \frac{\delta ^{2}S_{2}\left( \Psi \left( K,X,\theta \right) \right) }{\delta \Psi \left( K,X,\theta \right) \delta \Psi ^{\dag }\left( K^{\prime },X^{\prime },\theta ^{\prime }\right) }\Delta ^{\prime }\Psi \left( K^{\prime },X^{\prime },\theta ^{\prime }\right) \end{aligned}$$

by using that in average \(\left\langle \Delta \Psi \left( K,X,\theta \right) \Delta \Psi ^{\dag }\left( K^{\prime },X^{\prime },\theta ^{\prime }\right) \right\rangle \simeq 0\) for \(\theta \ne \theta ^{\prime }\). We are thus left with:

$$\begin{aligned} \int \Delta ^{\prime }\Psi ^{\dag } \left( K,X,\theta \right) \frac{\delta ^{2}S_{2}\left( \Psi \left( K,X,\theta \right) \right) }{\delta \Psi \left( K,X,\theta \right) \delta \Psi ^{\dag }\left( K^{\prime },X^{\prime },\theta ^{\prime }\right) }\Delta ^{\prime }\Psi \left( K,X,\theta \right) \end{aligned}$$

where the expression \(\frac{\delta S_{2}\left( \Psi \left( K,X,\theta \right) \right) }{\delta \Psi ^{\dag }\left( K,X,\theta \right) }\) has already been computed in (146) to find the fundamental state. The second-order derivative, in the notations of (146), is then:

$$\begin{aligned} \frac{\delta ^{2}S_{2}\left( \Psi \left( X\right) \right) }{\delta \Psi \left( X\right) \delta \Psi ^{\dag }\left( X\right) }= & {} -\frac{\sigma _{X}^{2} }{2}\nabla ^{2}+\frac{\kappa _{0}}{2\sigma _{X}^{2}}\left( X-\left\langle X\right\rangle \right) ^{2}+2V_{1}\left( \left| X-Y\right| \right) \left| \Psi \left( Y\right) \right| ^{2} \nonumber \\&+3V_{2}\left( \left| X-Y\right| ,\left| X-Z\right| ,\left| Y-Z\right| \right) \left| \Psi \left( Y\right) \right| ^{2}\left| \Psi \left( Z\right) \right| ^{2}\nonumber \\ \end{aligned}$$
(199)

This is the operator involved in the fundamental state equation. Using (152), we find the second-order term for the X-part of the action:

$$\begin{aligned}&\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( \frac{ \kappa _{0}}{2\sigma _{X}^{2}}\left( 1+\frac{2\sigma _{X}^{2}}{\kappa _{0}} \left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) \right) \right) \nonumber \\&\quad \times \left( X-\frac{sgn\left( X\right) \sigma _{X}^{2}}{ \kappa _{0}}\left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2}\Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\&\quad +\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( \rho ^{4}\kappa _{2}-\rho ^{2}\kappa _{1}-\frac{\sigma _{X}^{2}}{2\kappa _{0}} \left( \chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2}\right) \Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\ \end{aligned}$$
(200)

The norm \(\rho ^{2}\) of \(\Psi _{0}\) was derived at zeroth order in \(\chi _{1} \) and \(\chi _{2}\) in “Appendix 4”, Eq. (149):

$$\begin{aligned} \rho ^{2}=\frac{\kappa _{1}+\sqrt{\kappa _{1}^{2}-2\kappa _{2}\left( 2\alpha +\frac{1}{\vartheta ^{2}}+\sqrt{\kappa _{0}}+\sqrt{\delta ^{2}+\frac{\bar{A} ^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}\right) }}{2\kappa _{2} } \end{aligned}$$
(201)

for \(\bar{A}^{2}>>A^{2}\). The complete action is then obtained by gathering (200), (197) and (175):

$$\begin{aligned}&\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( -\frac{ \sigma ^{2}}{2}\nabla _{K}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}- \frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}+\frac{\omega ^{2}}{2\sigma ^{2}} \left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}+\alpha _{X}\right) \nonumber \\&\quad \times \Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\&\quad +\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \frac{\omega _{X}^{2}}{2\sigma _{X}^{2}}\left( X-\frac{sgn\left( X\right) \sigma _{X}^{2} }{\kappa _{0}}\left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2}\nonumber \\&\quad \times \Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\&\quad +\left( 2\kappa _{2}\rho ^{2}-\kappa _{1}\right) \rho ^{2}\left| \int \Delta \Psi \left( K,X,\theta \right) \Psi _{0}\left( K,X,\theta \right) \right| ^{2} \end{aligned}$$
(202)

where:

$$\begin{aligned} \omega= & {} \sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}} \nonumber \\ \omega _{X}= & {} \sqrt{\kappa _{0}\left( 1+\frac{2\sigma _{X}^{2}}{\kappa _{0}} \left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) \right) } \nonumber \\ \alpha _{X}= & {} \alpha +\frac{1}{2\vartheta ^{2}}+\left( -\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}-\frac{\sigma _{X}^{2}}{2\kappa _{0}}\left( \chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2}\right) \nonumber \\ \end{aligned}$$
(203)

Since (160) implies that:

$$\begin{aligned} 0= & {} \alpha +\frac{1}{2\vartheta ^{2}}+\frac{1}{2}\kappa _{0}^{\frac{1}{2}} \sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( -\frac{\kappa _{1}}{2} \chi _{1}^{2}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) } \\&-\kappa _{1}\rho ^{2}+\kappa _{2}\rho ^{4}-\frac{\sigma _{X}^{2}}{2\kappa _{0}}\left( \chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X,0}} \rho ^{2}-2\chi _{2}\kappa _{2}\rho ^{4}\right) ^{2}+\frac{\sqrt{\delta ^{2}+ \frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}{2} \end{aligned}$$

we have:

$$\begin{aligned} \alpha _{X}= & {} -\frac{1}{2}\kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\frac{K}{ \left\langle K\right\rangle _{X,0}}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }-\frac{\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}{2} \nonumber \\\simeq & {} -\frac{1}{2}\kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2} }{\kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }-\frac{\sqrt{\delta ^{2}+\frac{\bar{A} ^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}{2} \end{aligned}$$
(204)

and the second-order action (202) rewrites:

$$\begin{aligned}&\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( -\frac{ \sigma ^{2}}{2}\nabla _{K}^{2}-\frac{\vartheta ^{2}}{2}\nabla _{\theta }^{2}- \frac{\sigma _{X}^{2}}{2}\nabla _{X}^{2}+\frac{\omega ^{2}}{2\sigma ^{2}} \left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}\right) \Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\&\quad +\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \frac{\omega _{X}^{2}}{2\sigma _{X}^{2}}\left( X-\frac{sgn\left( X\right) \sigma _{X}^{2} }{\kappa _{0}}\left( -\chi _{1}\kappa _{1}\frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \right) ^{2} \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \nonumber \\&\quad -\int \Delta ^{\prime }\Psi ^{\dag }\left( K,X,\theta \right) \left( \frac{ 1}{2}\kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0}} \left( -\frac{\chi _{1}^{2}\kappa _{1}\rho ^{2}}{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) }+\frac{\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{ \left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}}{2}\right) \Delta ^{\prime }\Psi \left( K,X,\theta \right) \nonumber \\&\quad +\left( 2\kappa _{2}\rho ^{2}-\kappa _{1}\right) \rho ^{2}\left| \int \Delta \Psi \left( K,X,\theta \right) \Psi _{0}\left( K,X,\theta \right) \right| ^{2} \end{aligned}$$
(205)

Computation of the Green function The Green function is computed by including the influence of the background field. We have shown that the fundamental level of the X-part of the action has the form:

$$\begin{aligned} \Psi _{0}^{\left( 1\right) }\left( X\right)= & {} N_{1}\exp \left( -\frac{ \kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( - \frac{\kappa _{1}}{2}\chi _{1}^{2}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }}{2\sigma _{X}^{2}}\left( X-\delta X\right) ^{2}\right) H\left( X\right) \\&+N_{1}\exp \left( -\frac{\kappa _{0}^{\frac{1}{2}}\sqrt{1+\frac{2\sigma _{X}^{2}}{\kappa _{0}}\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\rho ^{2}+2\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) }}{2\sigma _{X}^{2}}\left( X+\delta X\right) ^{2}\right) H\left( -X\right) \\\simeq & {} N_{1}\exp \left( -\frac{\kappa _{0}^{\frac{1}{2}}\left( X-\delta X\right) ^{2}}{2\sigma _{X}^{2}}\right) H\left( X\right) +N_{1}\exp \left( - \frac{\kappa _{0}^{\frac{1}{2}}\left( X+\delta X\right) ^{2}}{2\sigma _{X}^{2}}\right) H\left( -X\right) \\\equiv & {} \bar{\Psi }_{0}^{\left( 1\right) }\left( X-\delta X\right) H\left( X\right) +\bar{\Psi }_{0}^{\left( 1\right) }\left( X+\delta X\right) H\left( -X\right) \end{aligned}$$

where:

$$\begin{aligned} \delta X=\frac{\sigma _{X}^{2}}{\kappa _{0}}\left( -\chi _{1}\kappa _{1} \frac{K}{\left\langle K\right\rangle _{X,0}}\rho ^{2}+2\chi _{2}\kappa _{2}\rho ^{4}\right) \end{aligned}$$

For a given value of K, the Green function for the X-part of the action is obtained by its expansion as a function of all the eigenstates \(\bar{\Psi } _{n}\) of the system:

$$\begin{aligned} G_{K}\left( X,X^{\prime }\right)= & {} \sum _{n}\left( \bar{\Psi }_{n}\left( X-\delta X\right) H\left( X\right) +\bar{\Psi }_{n}\left( X+\delta X\right) H\left( -X\right) \right) \\&\times \left( \bar{\Psi }_{n}\left( X^{\prime }-\delta X\right) H\left( X^{\prime }\right) +\bar{\Psi }_{n}\left( X^{\prime }+\delta X\right) H\left( -X^{\prime }\right) \right) \\= & {} \bar{G}\left( X-\delta X,X^{\prime }-\delta X\right) H\left( X\right) H\left( X^{\prime }\right) \\&+\bar{G}\left( X+\delta X,X^{\prime }+\delta X\right) H\left( -X\right) H\left( -X^{\prime }\right) \\&+\bar{G}\left( X-\delta X,X^{\prime }+\delta X\right) H\left( X\right) H\left( -X^{\prime }\right) \\&+\bar{G}\left( X+\delta X,X^{\prime }-\delta X\right) H\left( -X\right) H\left( X^{\prime }\right) \end{aligned}$$

where we have defined:

$$\begin{aligned}&\bar{G}\left( X,X^{\prime },\theta ,\theta ^{\prime }\right) \\&\quad =\sqrt{\frac{ \bar{\omega }_{X}}{2\pi \sigma _{X}^{2}\sinh \left( \bar{\omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) }} \\&\qquad \times \exp \left( \left( -\frac{\bar{\omega }_{X}}{2\sigma _{X}^{2}\sinh \left( \bar{\omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) } \right) \left( \left( X^{2}+\left( X^{\prime }\right) ^{2}\right) \right. \right. \\&\left. \left. \qquad \cosh \left( \bar{\omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) -2XX^{\prime }\right) \right) \\&\quad =\sqrt{\frac{\bar{\omega }_{X}}{2\pi \sigma _{X}^{2}\sinh \left( \bar{\omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) }} \\&\qquad \times \exp \left( \left( -\frac{\bar{\omega }_{X}}{2\sigma _{X}^{2}\sinh \left( \bar{\omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) } \right) \left( \left( X-X^{\prime }\right) ^{2}\right. \right. \\&\left. \left. \qquad +\left( \cosh \left( \bar{ \omega }_{X}\left( \theta ^{\prime }-\theta \right) \right) -1\right) \left( X^{2}+\left( X^{\prime }\right) ^{2}\right) \right) \right) \end{aligned}$$

with:

$$\begin{aligned} \bar{\omega }_{X}=\sqrt{\kappa _{0}\left( 1+\frac{\sigma _{X}^{2}}{\kappa _{0} }\left( -\frac{\kappa _{1}}{2}\chi _{1}^{2}\left( \frac{K}{\left\langle K\right\rangle _{X,0}}+\frac{K^{\prime }}{\left\langle K\right\rangle _{X^{\prime },0}}\right) \rho ^{2}+4\kappa _{2}\chi _{2}^{2}\rho ^{4}\right) \right) } \end{aligned}$$

As in the first phase, we have replaced \(\omega _{X}\) defined in (203) by \(\bar{\omega }_{X}\), its average over the trajectory.

The exponential term associated to the change of variable is computed as in the first phase. However, it has to be evaluated in the state \(\Psi _{0}\left( K,X,\theta \right) +\Psi \left( K,X,\theta \right) \). “Appendix 4” showed how, in first approximation, this amounts to computing it in the state \(\Psi _{0}\left( K,X,\theta \right) \). We find again an exponential factor:

$$\begin{aligned} \exp \left( -\left[ \delta \frac{\left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}}{2\sigma ^{2}}\right] _{\left( K,X\right) }^{\left( K^{\prime },X^{\prime }\right) }\right) \end{aligned}$$

with \(\left\langle K\right\rangle _{X}\) computed in the state \(\Psi _{0}\left( K,X,\theta \right) \), as in “Appendix 4”.

Ultimately, we can associate a Green function to (205), as in the first phase. It is the Laplace transform of a temporal Green function with parameter \(\alpha _{X}\) given in (204):

$$\begin{aligned}&G\left( K,K^{\prime },P,P^{\prime },X,X^{\prime },\theta ,\theta ^{\prime },t\right) \\&\quad =\exp \left( -\left[ \delta \frac{\left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}}{2\sigma ^{2}}\right] _{\left( K,X\right) }^{\left( K^{\prime },X^{\prime }\right) }\right) \\&\qquad \times \sqrt{\frac{\omega /2\pi \sigma ^{2}}{\sinh \left( \omega t\right) } }\exp \left( -\frac{\omega \left( \left( \left( K-\left\langle K\right\rangle \right) ^{2}+\left( K^{\prime }-\left\langle K\right\rangle \right) ^{2}\right) \cosh \left( \omega t\right) -2\left( K-\left\langle K\right\rangle \right) \left( K^{\prime }-\left\langle K\right\rangle \right) \right) }{2\sigma ^{2}\sinh \left( \omega t\right) }\right) \\&\qquad \times G_{\frac{K+K^{\prime }}{2}}\left( X,X^{\prime },\theta ,\theta ^{\prime }\right) \\&\qquad \times \sqrt{\frac{1}{2\pi \vartheta ^{2}t}}\exp \left( -\frac{\left( \theta ^{\prime }-\theta \right) ^{2}}{2\vartheta ^{2}t}+\frac{\theta ^{\prime }-\theta }{\vartheta ^{2}}\right) \times \delta \left( P-\frac{ D\exp \left( -\frac{\left| X\right| }{1+\gamma }\right) }{\left( \frac{K}{\left\langle K\right\rangle }\right) ^{\frac{\alpha }{1+\gamma }}} \right) \\&\qquad \times \delta \left( P^{\prime }-\frac{D\exp \left( -\frac{ \left| X^{\prime }\right| }{1+\gamma }\right) }{\left( \frac{ K^{\prime }}{\left\langle K\right\rangle }\right) ^{\frac{\alpha }{1+\gamma } }}\right) \end{aligned}$$

where, as in (195), we have used the average of U to compute \(\omega \):

$$\begin{aligned} \omega= & {} \sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}} \\ U= & {} 1-h\exp \left( -\frac{1}{d}\right) \left( 1-\frac{\cosh \frac{1}{2d}}{ \exp \left( \frac{1}{d}\right) }\right) \end{aligned}$$

Even though \(\alpha _{X}<0\) [see (204)], the Laplace transform is well defined, since \(-\alpha _{X}\) is the lower bound of the terms in the exponential.Footnote 9 The value of \(\omega \) is given by (194) with h defined in (143).

As in phase 1, this Green function is centred around \(t=\theta ^{\prime }-\theta \), so that we can replace \(t=\theta ^{\prime }-\theta \) in the Green function. This leads to:

$$\begin{aligned}&G\left( K,K^{\prime },P,P^{\prime },X,X^{\prime },\theta ,\theta ^{\prime }\right) \\&\quad =\exp \left( -\left[ \delta \frac{\left( K-\left\langle K\right\rangle _{X,0}\right) ^{2}}{2\sigma ^{2}}\right] _{\left( K,X\right) }^{\left( K^{\prime },X^{\prime }\right) }\right) \\&\qquad \times \sqrt{\frac{\omega }{2\pi \sigma ^{2}\sinh \left( \omega \left( \theta ^{\prime }-\theta \right) \right) }}\\&\qquad \exp \left( -\frac{\omega \left( \left( \left( K-\left\langle K\right\rangle \right) ^{2}+\left( K^{\prime }-\left\langle K\right\rangle \right) ^{2}\right) \cosh \left( \omega \left( \theta ^{\prime }-\theta \right) \right) -2\left( K-\left\langle K\right\rangle \right) \left( K^{\prime }-\left\langle K\right\rangle \right) \right) }{2\sigma ^{2}\sinh \left( \omega \left( \theta ^{\prime }-\theta \right) \right) }\right) \\&\qquad \times G_{\frac{K+K^{\prime }}{2}}\left( X,X^{\prime },\theta ,\theta ^{\prime }\right) \times \delta \left( P-\frac{D\exp \left( -\frac{ \left| X\right| }{1+\gamma }\right) }{\left( \frac{K}{\left\langle K\right\rangle }\right) ^{\frac{\alpha }{1+\gamma }}}\right) \\&\qquad \times \delta \left( P^{\prime }-\frac{D\exp \left( -\frac{\left| X^{\prime }\right| }{1+\gamma }\right) }{\left( \frac{K^{\prime }}{\left\langle K\right\rangle }\right) ^{\frac{\alpha }{1+\gamma }}}\right) H\left( \theta ^{\prime }-\theta \right) \end{aligned}$$

As in the case \(\rho =0\), we can replace \(\bar{\omega }\) by \(\omega \) to obtain the formula in the text.

Appendix 6

This appendix studies the emergence of a K-dependent barrier potential as described in section 5.4.2 of the text. To do so, we consider a general model:

$$\begin{aligned} S\left( \Psi \right) =\int \Psi ^{\dag }\left( X\right) \left( -\sigma _{X}^{2}\nabla _{X}^{2}+V\right) \Psi \left( X\right) +\int \frac{1}{2}\Psi ^{\dag }\left( X\right) \Psi ^{\dag }\left( Y\right) W\left( X,Y\right) \Psi \left( X\right) \Psi \left( Y\right) \end{aligned}$$

that encompasses the model studied in this paper. The field \(\Psi \left( X\right) \) depends on an arbitrary number of variables X belonging to some configuration space, and \(W\left( X,Y\right) =W\left( Y,X\right) \). We have chosen a fourth-order interaction term, but a more general choice, such as a sum of powers, would not change the result. We assume that there is a non-trivial minimum to the action \(S\left( \Psi \right) \), so that the equation:

$$\begin{aligned} \left( -\sigma _{X}^{2}\nabla _{X}^{2}+V\right) \Psi \left( X\right) +\left( \int \Psi ^{\dag }\left( Y\right) W\left( X,Y\right) \Psi \left( Y\right) \right) \Psi \left( X\right) =0 \end{aligned}$$

has a solution \(\rho \Psi _{0}\left( X\right) \ne 0\), and \(\Psi _{0}\) of norm equal to 1. We show that the non-trivial vacuum implies separating the system into two sub-systems defined on two half-space of the configuration space.

Given that the field \(\Psi _{0}\) minimises the action, the second-order variation of \(S\left( \Psi \right) \) is:

$$\begin{aligned}&\int \Delta \Psi ^{\dag }\left( X\right) \left( -\sigma _{X}^{2}\nabla _{X}^{2}+V\right) \Delta \Psi \left( X\right) \hbox {d}X+\Delta \Psi ^{\dag }\left( X\right) \nonumber \\&\qquad \left( \int \rho ^{2}\Psi _{0}^{\dag }\left( Y\right) W\left( X,Y\right) \Psi _{0}\left( Y\right) \hbox {d}Y\right) \Delta \Psi \left( X\right) \hbox {d}X \nonumber \\&\qquad +2\int \mathfrak {Re}\left( \Delta \Psi ^{\dag }\left( X\right) \left( \rho ^{2}\Psi _{0}^{\dag }\left( Y\right) W\left( X,Y\right) \Psi _{0}\left( X\right) \right) \Delta \Psi \left( Y\right) \right) \hbox {d}X\hbox {d}Y \nonumber \\&\quad =\int \Delta \Psi ^{\dag }\left( X\right) \left( \left( -\sigma _{X}^{2}\nabla _{X}^{2}+V\left( X\right) \right) \Delta \Psi \left( X\right) +\Delta \Psi ^{\dag }\left( X\right) V_{0}\left( X\right) \Delta \Psi \left( X\right) \right) \hbox {d}X \nonumber \\&\qquad +\int \Delta \Psi ^{\dag }\left( X\right) \left( W_{0}\left( X,Y\right) +W_{0}^{\dag }\left( X,Y\right) \right) \Delta \Psi \left( Y\right) \hbox {d}X\hbox {d}Y \end{aligned}$$
(206)

where:

$$\begin{aligned} W_{0}\left( Y,X\right)= & {} \rho ^{2}\Psi _{0}^{\dag }\left( Y\right) W\left( X,Y\right) \Psi _{0}\left( X\right) \\ W_{0}^{\dag }\left( X,Y\right)= & {} W_{0}\left( Y,X\right) \end{aligned}$$

and:

$$\begin{aligned} V_{0}\left( X\right) =\int \rho ^{2}\Psi _{0}^{\dag }\left( Y\right) W\left( X,Y\right) \Psi _{0}\left( Y\right) \end{aligned}$$

Now, assume that \(\Psi _{0}^{\left( 1\right) }\left( X\right) \) is peaked around some \(X_{0}\), which is the case for equation (60) in the text for the K-part of \(\Psi _{0}\) (see (43)). Then:

$$\begin{aligned} V_{0}\left( X\right)\simeq & {} \int \rho ^{2}\Psi _{0}^{\dag }\left( Y\right) W\left( X,X_{0}\right) \Psi _{0}\left( Y\right) =W\left( X,X_{0}\right) \rho ^{2} \\ W_{0}\left( Y,X\right)\simeq & {} \rho ^{2}\int \Psi _{0}^{\dag }\left( X_{0}\right) W\left( X_{0},X_{0}\right) \Psi _{0}\left( X\right) \delta \left( X-X_{0}\right) \delta \left( Y-X_{0}\right) \end{aligned}$$

and the following contributions of the second-order variation become:

$$\begin{aligned}&\Delta \Psi ^{\dag }\left( X\right) V_{0}\left( X\right) \Delta \Psi \left( X\right) \rightarrow \rho ^{2}\Delta \Psi ^{\dag }\left( X\right) W\left( X,X_{0}\right) \Delta \Psi \left( X\right) \\&\Delta \Psi ^{\dag }\left( X\right) \left( W_{0}\left( X,Y\right) +W_{0}^{\dag }\left( X,Y\right) \right) \Delta \Psi \left( Y\right) \rightarrow 2\rho ^{2}\left| \Psi _{0}\left( X\right) \right| ^{2}W\left( X_{0},X_{0}\right) \Delta \Psi ^{\dag }\left( X_{0}\right) \Delta \Psi \left( X_{0}\right) \end{aligned}$$

For \(\rho ^{2}W\left( X_{0},X_{0}\right)>>1\), the contributions of the statistical weight due to the fields \(\Delta \Psi \) such that \(\Delta \Psi \left( X_{0}\right) \ne 0\) are suppressed. Then the integrals over \(\Delta \Psi \) can be limited to contributions such that \(\Delta \Psi \left( X_{0}\right) =0\).

This means that \(\Delta \Psi \) can be decomposed into two parts:

$$\begin{aligned} \Delta \Psi \left( X\right) =\Delta \Psi _{+}\left( X\right) +\Delta \Psi _{-}\left( X\right) \end{aligned}$$

where \(\Delta \Psi _{+}\left( X\right) \) and \(\Delta \Psi _{-}\left( X\right) \) are independent and defined on two half space \(X_{\pm }\), respectively. They satisfy:

$$\begin{aligned} \Delta \Psi _{\pm }\left( X_{\mp }\right) =0 \end{aligned}$$

As a consequence, the action at the second order for \(\Delta \Psi _{\pm }\) becomes:

$$\begin{aligned} \Delta \Psi _{\pm }^{\dag }\left( X\right) \left( -\sigma _{X}^{2}\nabla _{X}^{2}+V\left( X\right) \right) \Delta \Psi _{\pm }\left( X\right) +\rho ^{2}\Delta \Psi _{\pm }^{\dag }\left( X\right) W\left( X,X_{0}\right) \Delta \Psi _{\pm }\left( X\right) \end{aligned}$$

This action also models two independent fields with constraint \(\Delta \Psi _{\pm }\left( X_{\mp }\right) =0\). They are defined on the whole space, but constrained by a potential wall \(H_{\pm }\left( X\right) \), this wall being defined on the space \(X_{\mp }\). The second-order action (206) becomes:

$$\begin{aligned} \Delta \Psi _{\pm }^{\dag }\left( X\right) \left( -\sigma _{X}^{2}\nabla _{X}^{2}+V\left( X\right) +H_{\pm }\left( X\right) \right) \Delta \Psi _{\pm }\left( X\right) +\rho ^{2}\Delta \Psi _{\pm }^{\dag }\left( X\right) W\left( X,X_{0}\right) \Delta \Psi _{\pm }\left( X\right) \end{aligned}$$

This models two sets of different agents, evolving on \(X_{\pm }\) and subject to a wall potential. Applied to our case, the results are the following. Recall that we found the fundamental state (157):

$$\begin{aligned} \Psi _{0}\left( K,X\right)= & {} \rho N\left[ \bar{\Psi }_{0}^{\left( 1\right) }\left( X-\delta X\right) H\left( X\right) +\bar{\Psi }_{0}^{\left( 1\right) }\left( X+\delta X\right) H\left( -X\right) \right] \\&\times \exp \left( -\frac{\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A}^{2}\right) ^{2}}}\left( K-\left\langle K\right\rangle _{X}\right) ^{2}}{2\sigma }\right) \end{aligned}$$

where N and \(\rho \) are given by (158) and (149) and (159), respectively. If we assume, as we did in “Appendix 3”, that:

$$\begin{aligned} \frac{\sqrt{\delta ^{2}+\frac{\bar{A}^{2}A^{2}}{\left( A^{2}U^{2}+\bar{A} ^{2}\right) ^{2}}}}{\sigma }>\kappa _{0}^{\frac{1}{2}} \end{aligned}$$

i.e. that the X-variable is more spread than K, then \(\Psi _{0}\left( K,X\right) \) is peaked on the hypersurface \(K=\left\langle K\right\rangle _{X}\). As a consequence, the space \(\left( K,X\right) \) is divided into two subspaces, \(S_{+}\), defined by \(K>\left\langle K\right\rangle _{X}\), and \(S_{-}\), defined by \(K<\left\langle K\right\rangle _{X}\). These half-spaces correspond to two systems that are independent in first approximation. An agent starting in \(S_{+}\) (\(S_{-}\), respectively) will remain in \(S_{+}\) (\( S_{-}\), respectively).

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Gosselin, P., Lotz, A. & Wambst, M. A statistical field approach to capital accumulation. J Econ Interact Coord 16, 817–908 (2021). https://doi.org/10.1007/s11403-021-00330-9

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Keywords

  • Path integrals
  • Statistical field theory
  • Phase transition
  • Capital accumulation
  • Exchange space
  • Multi-agent model
  • Interaction agents

JEL Classification

  • C02
  • C60
  • E00
  • E1