Minimal Contagious Sets in Random Regular Graphs

Abstract

The bootstrap percolation (or threshold model) is a dynamic process modelling the propagation of an epidemic on a graph, where inactive vertices become active if their number of active neighbours reach some threshold. We study an optimization problem related to it, namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics, with and without a constraint on the time needed for the complete activation. This problem encompasses in special cases many extremal characteristics of graphs like their independence, decycling or domination number, and can also be seen as a packing problem of repulsive particles. We use the cavity method (including the effects of replica symmetry breaking), an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years. We have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs, the most striking being that 5-regular random graph with a threshold of activation of 3 (resp. 6-regular with threshold 4) have contagious sets containing a fraction $1/6$ (resp. $1/4$) of the total number of vertices. Equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5-regular (resp. 6-regular) random graph to destroy its 3-core. We also investigated Survey Propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs.

Appendices

Appendix 1: The Limit $\mu \rightarrow -\infty$ of the Fields Recursion

We justify here the Eq. (73) for the recursion $h=g(h_1,\ldots ,h_k)$ between “hard fields” $h_i\in \{A_0,A_1,\ldots ,A_{T-1},A_T=B_T,B_{T-1},\ldots ,B_1,B_0 \}$. We can first notice that in Eqs. (70, 71) the (constrained) maximum over the partitions $I,J,K$ of $\mathcal{S}_t$ is always reached for $|I|+|J|$ and $|I|$ as small as possible (because $a_t^{(i)} \ge b_{t-1}^{(i)} \ge b_{t-2}^{(i)}$), which allows to rewrite

$$\begin{aligned} a_t&= \max \left( 0, \max _{t^{\prime }\in [1,T]} \max _{\begin{array}{c} J,K \\ |J| = l - {\mathbb {I}}(t^{\prime } \ge t+1) \end{array} } \mathcal{S}_{t^{\prime }}(h_1,\ldots ,h_k;\emptyset ,J,K) \right) , \end{aligned}$$

(106)

$$\begin{aligned} b_t&= \max \left( 0, \max _{t^{\prime }\in [1,t]} \max _{\begin{array}{c} J,K \\ |J| = l \end{array}} \mathcal{S}_{t^{\prime }}(h_1,\ldots ,h_k;\emptyset ,J,K) \right) , \end{aligned}$$

(107)

where $J,K$ forms a partition of $\{1,\ldots ,k\}$. In addition one realizes that

$$\begin{aligned}&\max _{\begin{array}{c} J,K \\ |J| = l \end{array}} \mathcal{S}_t (h_1,\ldots ,h_k;\emptyset ,J,K) = 1 \nonumber \\&\Leftrightarrow \left( \sum _{i=1}^k {\mathbb {I}}(h_i \in \{A_0,\ldots ,A_{t-1} \} ) \!=\! 0 \ \ \text {and} \ \ \sum _{i=1}^k {\mathbb {I}}(h_i \in \{B_0,\ldots ,B_{t-1} \} ) \ge l \right) \!, \end{aligned}$$

(108)

which by logical negation leads to

$$\begin{aligned}&\max _{\begin{array}{c} J,K \\ |J| = l \end{array}} \mathcal{S}_t (h_1,\ldots ,h_k;\emptyset ,J,K) \le 0 \nonumber \\&\Leftrightarrow \left( \sum _{i=1}^k {\mathbb {I}}(h_i \in \{A_0,\ldots ,A_{t-1} \} ) \ge 1 \ \ \text {or} \ \ \sum _{i=1}^k {\mathbb {I}}(h_i \in \{B_0,\ldots ,B_{t-1} \} ) \le l-1 \right) \!. \end{aligned}$$

(109)

Combining these logical rules leads after a short reasoning to

$$\begin{aligned} g(h_1,\ldots ,h_k)=A_t&\Leftrightarrow \left( a_t =1 \ \text {and} \ a_{t+1}=0 \right) \end{aligned}$$

(110)

$$\begin{aligned}&\Leftrightarrow {\left\{ \begin{array}{ll} &{} \sum _{i=1}^k {\mathbb {I}}(h_i \in \{B_0,\ldots ,B_t \}) = l-1 \\ \text {and}&{} \sum _{i=1}^k {\mathbb {I}}(h_i \in \{A_0,\ldots ,A_t \})=0 \\ \text {and}&{} \sum _{i=1}^k {\mathbb {I}}(h_i=A_{t+1}) \ge 1 \end{array}\right. }\!, \end{aligned}$$

(111)

and

$$\begin{aligned} g(h_1,\ldots ,h_k)=B_t&\Leftrightarrow \left( b_t =1 \ \text {and} \ b_{t-1}=0 \right) \end{aligned}$$

(112)

$$\begin{aligned}&\Leftrightarrow {\left\{ \begin{array}{ll} &{} \sum _{i=1}^k {\mathbb {I}}(h_i \in \{B_0,\ldots ,B_{t-1} \}) \ge l \\ \text {and}&{} \sum _{i=1}^k {\mathbb {I}}(h_i \in \{B_0,\ldots ,B_{t-2} \})\le l-1 \\ \text {and}&{} \sum _{i=1}^k {\mathbb {I}}(h_i \in \{A_0,\ldots ,A_{t-1} \})=0 \end{array}\right. }\!. \end{aligned}$$

(113)

Considering the various possible cases leading to a field of type $A_t$ or $B_t$ yields finally (73).

Appendix 2: Technical Details on the Resolution of the Factorized RS and Energetic 1RSB Equations

We shall present in this Appendix the details of the RS and energetic 1RSB cavity equations in the particular case of random $k+1$ regular graphs with an uniform threshold $l$ of activations. It turns out that despite their different interpretations these two version of the cavity method can be treated in an unified way. We thus begin by introducing this common formulation, then we unveil the simplifications that arise in the case $l=k$, before finally discussing the limit $T\rightarrow \infty$, both in the case $l=k$ and $l<k$.

1.1 Common Formulation

1.1.1 RS Cavity Method

Consider the fixed-point RS equation $h=g(h,\ldots ,h)$, with $g$ defined in Eq. (40); alternatively we saw in Eqs. (66, 67) an expression for the differences $e^{-\mu a_t} - e^{-\mu a_{t+1}}$. Setting $h_i=h$ in the right-hand sides of these equations, and using the identity

$$\begin{aligned} \sum _{\begin{array}{c} I,J,K \\ |I| \le l-1 \\ |I|+|J| \ge l \end{array}} f(I,J,K) = \sum _{\begin{array}{c} I,J,K \\ |I|+|J| \ge l \end{array}} f(I,J,K) - \sum _{\begin{array}{c} I,J,K \\ |I| \ge l \end{array}} f(I,J,K) \ , \end{aligned}$$

(114)

for any function $f$ of a partition $I,J,K$, allows to show the equivalence of the fixed-point equation on $h=(a_0,\ldots ,a_T,b_{T-1},\ldots ,b_1)$ with:

$$\begin{aligned} e^{-\mu a_t}- e^{-\mu a_{t+1}}&= e^{-\mu + \mu k a_0} \left( {\begin{array}{c}k\\ l-1\end{array}}\right) e^{-\mu (l-1) b_t}\nonumber \\&\times \left[ \left( e^{-\mu a_{t+1}} \!-\! e^{-\mu b_t } \right) ^{k-l+1} \!-\! \left( e^{-\mu a_{t+2}} - e^{-\mu b_t } \right) ^{k-l+1} \right] \ ,\end{aligned}$$

(115)

$$\begin{aligned} e^{-\mu b_{t+1}}- e^{-\mu b_t}&= e^{-\mu + \mu k a_0} \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \nonumber \\&\times \left[ e^{-\mu p b_t} \left( e^{-\mu a_{t+1}} \!-\! e^{-\mu b_t} \right) ^{k-p} \!-\! e^{-\mu p b_{t-1}} \left( e^{-\mu a_{t+1}} - e^{-\mu b_{t-1}} \right) ^{k-p} \right] \!.\nonumber \\ \end{aligned}$$

(116)

These equations are valid for $t\in \{0,\ldots ,T-1\}$, with the boundary conditions $e^{-\mu b_{-1}}=0$, $b_0=1$, $a_T=b_T$, $a_{T+1}=b_{T-1}$. The thermodynamic quantities can also be simplified in this factorized case, the site contribution to the RS free-entropy reading from Eq. (43):

$$\begin{aligned} {z_\mathrm{site}}&= 1 + e^{-\mu + \mu (k+1) a_0} \sum _{t=1}^T \sum _{p=l}^{k+1} \left( {\begin{array}{c}k+1\\ p\end{array}}\right) \nonumber \\&\left[ e^{-\mu p b_{t-1}} \left( e^{-\mu a_t} - e^{-\mu b_{t-1}} \right) ^{k+1-p} - e^{-\mu p b_{t-2}} \left( e^{-\mu a_t} - e^{-\mu b_{t-2}} \right) ^{k+1-p} \right] \!, \end{aligned}$$

(117)

while the edge contribution of Eq. (42) becomes

$$\begin{aligned} {z_\mathrm{edge}}= e^{2 \mu a_0} \left[ e^{- 2 \mu a_T} + 2 \sum _{t=0}^{T-1} \left( e^{-\mu a_t} - e^{-\mu a_{t+1}}\right) e^{-\mu b_t} \right] . \end{aligned}$$

(118)

Let us introduce some new notations and define a change of parameters on the unknowns $a_t,b_t$, as $u_t=e^{-\mu a_t}$, $v_t=e^{-\mu b_t}$. We also define a new parameter $\lambda$, with $\lambda =e^{-\mu + \mu k a_0}$. In terms of these new quantities the above set of equations becomes

$$\begin{aligned} u_t - u_{t+1}&= D(u_{t+1},v_t) - D(u_{t+2},v_t) \ , \end{aligned}$$

(119)

$$\begin{aligned} v_{t+1} - v_t&= S(u_{t+1},v_t) - S(u_{t+1},v_{t-1}) \ , \end{aligned}$$

(120)

with $v_{-1}=0$, $v_0=1$, $u_T=v_T$, $u_{T+1}=v_{T-1}$, and

$$\begin{aligned} D(u,v) = \lambda \left( {\begin{array}{c}k\\ l-1\end{array}}\right) v^{l-1} (u-v)^{k-l+1} \ , \qquad S(u,v) = \lambda \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) v^p (u-v)^{k-p}\!. \end{aligned}$$

(121)

In other words the $u$’s and $v$’s are solutions of a set of polynomial equations, and as such should be viewed as a function of $\lambda$ and $T$ (and of course of $k$ and $l$). They also obey, on top of the boundary conditions, the inequalities $u_0 \ge u_1 \ge \cdots \ge u_T = v_T \ge v_{T-1} \ge \cdots v_1 \ge v_0=1$. The chemical potential $\mu$ has disappeared from this set of equations, but actually it is now implicitly a function of $\lambda$ and $T$, as from the definition of $\lambda$ one recovers $\mu$ with $\mu = -\ln (\lambda u_0^k)$.

For future use we emphasize here an identity between the derivatives of $D$ and $S$ and introduce a new function $C(u,v)$:

$$\begin{aligned} C(u,v) = \frac{\partial D}{\partial u} = \frac{\partial S}{\partial v} = \lambda l \left( {\begin{array}{c}k\\ l\end{array}}\right) v^{l-1} (u-v)^{k-l} \ . \end{aligned}$$

(122)

Let us also rewrite the thermodynamic quantities in terms of these new variables. The expressions (117) and (118) become

$$\begin{aligned} {z_\mathrm{site}}= 1 + {F_\mathrm{site}}\ , \qquad {z_\mathrm{edge}}= \frac{1}{u_0} {F_\mathrm{edge}}\ , \end{aligned}$$

(123)

where we introduced the two functions

$$\begin{aligned} {F_\mathrm{site}}(\lambda ,T) \!&= \! \frac{\lambda }{u_0} \sum _{t=1}^T \sum _{p=l}^{k+1} \left( {\begin{array}{c}k+1\\ p\end{array}}\right) \left[ v_{t-1}^p (u_t \!-\! v_{t-1})^{k+1-p} \!-\! v_{t-2}^p (u_t \!-\! v_{t-2})^{k+1-p} \right] \!,\end{aligned}$$

(124)

$$\begin{aligned} {F_\mathrm{edge}}(\lambda ,T) \!&= \! \frac{1}{u_0} \left[ v_T^2 \!+\! 2 \sum _{t=0}^{T-1} (u_t \!-\!u_{t+1}) v_t \right] \!. \end{aligned}$$

(125)

We emphasize here the dependency on $\lambda$ and $T$, which was kept implicit in the $u_t$ and $v_t$’s. One has then the final expressions of all RS thermodynamic quantities as:

$$\begin{aligned} \phi = \mu + \ln ({z_\mathrm{site}}) - \frac{k+1}{2} \ln ({z_\mathrm{edge}}) , \quad \mu = -\ln (\lambda u_0^k), \quad s = \phi - \mu \theta , \quad \theta = \frac{1}{{z_\mathrm{site}}}\!. \end{aligned}$$

(126)

One can also express the probability distribution of the activation times in terms of these new variables. Denoting $P_t$ the cumulative distribution, i.e. the probability that the activation time of one vertex is smaller or equal than $t$, one has from Eq. (45):

$$\begin{aligned} P_t = \frac{1}{{z_\mathrm{site}}}\left[ 1 + {F_\mathrm{site}}(\lambda ,T,t) \right] \ , \end{aligned}$$

(127)

where we defined

$$\begin{aligned} {F_\mathrm{site}}(\lambda ,T,t) =\frac{\lambda }{u_0} \sum _{t^{\prime }=1}^t \sum _{p=l}^{k+1} \left( {\begin{array}{c}k+1\\ p\end{array}}\right) \left[ v_{t^{\prime }-1}^p (u_{t^{\prime }} \!-\! v_{t^{\prime }-1})^{k+1-p} - v_{t^{\prime }-2}^p (u_{t^{\prime }} - v_{t^{\prime }-2})^{k+1-p}\right] \!. \nonumber \\ \end{aligned}$$

(128)

One can check that, as it should, $P_0=\theta$ the fraction of initially active sites (summations over empty sets being equal to zero by convention), and $P_T=1$ (as $\epsilon =+\infty$ all vertices are active at the final time).

1.1.2 Energetic 1RSB Cavity Method

We now turn to a similar study of the energetic 1RSB equations in the factorized case, namely the determination of the normalized vector of probabilities $P=(p_0,\ldots ,p_{T-1},q_T,\ldots ,q_0)$, solution of the fixed-point equation $P=G(P,\ldots ,P)$, with the mapping $G$ defined in Eq. (75).

Let us first note that in general the normalization $Z[P_1,\ldots ,P_k]$ of (75) can be expressed in terms of $q_0$,

$$\begin{aligned} Z = 1 + (e^y-1) (1-Z q_0) \ \Rightarrow \ \ \frac{e^y}{Z} = 1 + q_0 (e^y -1) \ . \end{aligned}$$

(129)

This remark allows to rewrite the fixed-point equation $P=G(P,\ldots ,P)$ as

$$\begin{aligned} p_t&= (1 + q_0 (e^y -1)) \left( {\begin{array}{c}k\\ l-1\end{array}}\right) \left( \sum _{t^{\prime }=0}^t q_{t^{\prime }}\right) ^{l-1}\\&\times \left[ \left( \sum _{t^{\prime }=t+1}^T q_{t^{\prime }} + \sum _{t^{\prime }=t+1}^{T-1} p_{t^{\prime }}\right) ^{k-l+1} - \left( \sum _{t^{\prime }=t+1}^T q_{t^{\prime }} + \sum _{t^{\prime }=t+2}^{T-1} p_{t^{\prime }}\right) ^{k-l+1} \right] \!,\\ q_t&= (1 + q_0 (e^y -1)) \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \\&\times \left[ \left( \sum _{t^{\prime }=0}^{t-1} q_{t^{\prime }}\right) ^p \left( \sum _{t^{\prime }=t}^T q_{t^{\prime }} + \sum _{t^{\prime }=t}^{T-1} p_{t^{\prime }}\right) ^{k-p} - \left( \sum _{t^{\prime }=0}^{t-2} q_{t^{\prime }}\right) ^p \left( \sum _{t^{\prime }=t-1}^T q_{t^{\prime }} + \sum _{t^{\prime }=t}^{T-1} p_{t^{\prime }}\right) ^{k-p} \right] \!, \end{aligned}$$

where in the first line $t\in \{0,\ldots ,T-1\}$ and in the second $t\in \{1,\ldots ,T\}$. These two sets of equations are supplemented by the normalization condition $q_0+\cdots +q_T+p_{T-1}+\cdots +p_0=1$.

The site and edge contributions of the energetic 1RSB potential, defined in (77, 79), become in the factorized case:

$$\begin{aligned} {\mathcal{Z}_\mathrm{site}}&= 1+(e^y-1)\sum _{t=1}^T \sum _{p=l}^{k+1} \left( {\begin{array}{c}k+1\\ p\end{array}}\right) \\&\times \left[ \left( \sum _{t^{\prime }=0}^{t-1} q_{t^{\prime }}\right) ^p \left( \sum _{t^{\prime }=t}^T q_{t^{\prime }} + \sum _{t^{\prime }=t}^{T-1} p_{t^{\prime }}\right) ^{k+1-p}\ - \left( \sum _{t^{\prime }=0}^{t-2} q_{t^{\prime }}\right) ^p \left( \sum _{t^{\prime }=t-1}^T q_{t^{\prime }} + \sum _{t^{\prime }=t}^{T-1} p_{t^{\prime }}\right) ^{k+1-p} \right] \!,\\ {\mathcal{Z}_\mathrm{edge}}&= e^{-y}+(1-e^{-y}) \left[ \left( \sum _{t=0}^T q_t \right) ^2 + 2 \sum _{t=0}^{T-1} p_t \sum _{t^{\prime }=0}^t q_{t^{\prime }} \right] \!. \end{aligned}$$

Now let us change variables and trade the unknowns $p_t,q_t$ for some variables $u_t$, $v_t$, and the parameter $y$ for some parameter $\lambda$, according to

$$\begin{aligned} u_t = \frac{1}{q_0} \left( \sum _{t^{\prime }=0}^T q_{t^{\prime }} + \sum _{t^{\prime }=t}^{T-1} p_{t^{\prime }} \right) \ , \quad v_t = \frac{1}{q_0} \sum _{t^{\prime }=0}^t q_{t^{\prime }} \ , \quad \lambda = (1 + q_0 (e^y -1)) q_0^{k-1} \ . \end{aligned}$$

(130)

Inserting these definitions in the above equations one realizes that the quantities $u_t$ and $v_t$ are solutions of exactly the same set of Eqs. (119, 120) defined in the RS case, and obey the same boundary conditions and inequalities. From the solution of these equations, for a given value of the parameter $\lambda$, one recovers the parameter $y$ noting that by the normalization condition one has $u_0=1/q_0$, hence $y = \ln (\lambda u_0^k - u_0 +1)$. The expressions of ${\mathcal{Z}_\mathrm{site}}$ and ${\mathcal{Z}_\mathrm{edge}}$ within this parametrization are easily obtained from the above equations and read:

$$\begin{aligned} {\mathcal{Z}_\mathrm{site}}= 1 + \left( 1 - \frac{1}{\lambda u_0^{k-1}} \right) {F_\mathrm{site}}\ , \qquad {\mathcal{Z}_\mathrm{edge}}= \frac{1+ (\lambda u_0^{k-1} - 1) {F_\mathrm{edge}}}{\lambda u_0^k - u_0 +1} \ , \end{aligned}$$

(131)

with the same functions ${F_\mathrm{site}}$ and ${F_\mathrm{edge}}$ defined in Eqs. (124, 125) for the RS case. One has finally an expression for the thermodynamic quantities of the energetic 1RSB formalism as

$$\begin{aligned} \Phi _\mathrm{e} = - y + \ln {\mathcal{Z}_\mathrm{site}}- \frac{k+1}{2} \ln {\mathcal{Z}_\mathrm{edge}}, \quad y = \ln (\lambda u_0^k - u_0 +1), \quad \Sigma _\mathrm{e} = \Phi _\mathrm{e} + y \theta \!, \end{aligned}$$

(132)

where $\theta$ is here the opposite of the derivative of $\Phi _\mathrm{e}$ with respect to $y$, which after a short computation reads

$$\begin{aligned} \theta&= 1- \frac{e^y}{e^y-1} \frac{{\mathcal{Z}_\mathrm{site}}- 1}{{\mathcal{Z}_\mathrm{site}}} - \frac{k+1}{2} \frac{1}{e^y-1} \frac{1-{\mathcal{Z}_\mathrm{edge}}}{{\mathcal{Z}_\mathrm{edge}}} \\&= \frac{1-\frac{1}{\lambda u_0^k} {F_\mathrm{site}}}{1 + \left( 1 - \frac{1}{\lambda u_0^{k-1}} \right) {F_\mathrm{site}}} - \frac{k+1}{2} \frac{1-\frac{1}{u_0} {F_\mathrm{edge}}}{1+(\lambda u_0^{k-1}-1) {F_\mathrm{edge}}} \ . \nonumber \end{aligned}$$

(133)

1.1.3 Simplifications for $l=k$

In the case $l=k$ further simplifications arise. Indeed the function $S(u,v)$ defined in (121) is in this case independent of $u$, and the Eqs. (119, 120) can be rewritten as:

$$\begin{aligned} v_0&= 1 \ , \end{aligned}$$

(134)

$$\begin{aligned} v_t&= 1 + \lambda \, v_{t-1}^k \qquad \text {for} \ t \in \{1,\ldots ,T\} \ , \end{aligned}$$

(135)

$$\begin{aligned} u_{T-1}&= v_T + \lambda k \, v_{T-1}^{k-1} \, (v_T - v_{T-1}) \ , \end{aligned}$$

(136)

$$\begin{aligned} u_t&= u_{t+1} + \lambda k \, v_t^{k-1} \, (u_{t+1} - u_{t+2}) \qquad \text {for} \ t \in \{0,\ldots ,T-2\}. \end{aligned}$$

(137)

This set of equations is particularly simple to solve, and admits a single solution for each value of $\lambda$. One can indeed compute by recurrence the value of the $v_t$ for increasing values of $t$ from $0$ to $T$, then deduce the value of $u_{T-1}$, and finally by a downward recurrence the values of $u_t$ for $t$ from $T-2$ to $0$. The thermodynamic observables are then deduced from (126) in the RS case or (132) in the energetic 1RSB case, where the site contributions can be simplified from (124), yielding

$$\begin{aligned} {F_\mathrm{site}}(\lambda ,T)=\frac{\lambda }{u_0}\left[ v_{T-1}^{k+1} + (k+1) \sum _{t=1}^T (u_t-u_{t+1}) v_{t-1}^k \right] . \end{aligned}$$

(138)

These simplifications can also be performed for the function (128) giving the distribution of activation times, which reads in the case $k=l$:

$$\begin{aligned} {F_\mathrm{site}}(\lambda ,T,t)\!=\!\frac{\lambda }{u_0}\left[ v_{t-1}^{k+1} \!+\! (k+1) v_{t-1}^k (u_{t+1}-v_{t-1})\!+\!(k+1) \sum _{t^{\prime }=1}^t (u_{t^{\prime }}-u_{t^{\prime }+1}) v_{t^{\prime }-1}^k \right] \!.\nonumber \\ \end{aligned}$$

(139)

1.1.4 Numerical Resolution for $l<k$

In the case $l<k$ we did not find a simple change of variables on the unknowns $u_t,v_t$ that would put the system of Eqs. (119, 120) in the triangular form that appeared naturally when $k=l$ and led to a direct resolution by successive substitutions. We therefore resorted to the Newton-Raphson iterative method for solving (119, 120), taking care of choosing a good initial condition for the iterations to be convergent. This guess on the solution was provided by analytical asymptotic expansions, either in the limit $\lambda \rightarrow 0$ or with $T\rightarrow \infty$ (see next paragraph). Depending on the values of $\lambda$ and $T$ we found either 0, 1 or 2 relevant solutions of (119, 120), but this multi valuedness has no physical meaning and comes only from the arbitrary choice of the parametrization in terms of $\lambda$. Indeed there is a single solution for each value of the chemical potential $\mu$ (or $y$ in the energetic 1RSB formalism).

1.2 The Large $T$ Limit

In the rest of this Appendix we shall justify analytically the claims made in Sects. 4.2.1 and 4.2.2 on the behaviour of the RS and energetic 1RSB solutions as $T$ goes to infinity.

1.2.1 The Trivial Solution

As anticipated in Sect. 4, in the large $T$ limit the portion of the curve $s(\theta )$ corresponding to $\theta >{\theta _\mathrm{r}}$ should coincide with the entropy $-\theta \ln \theta - (1-\theta ) \ln (1-\theta )$ counting all configurations with a fraction $\theta$ of initially active sites, as such configurations are typically activating (see the reminder on random initial configurations of Sect. 2.2). Let us see how to prove this statement. A moment of thought, considering for instance the form of the RS equations at $\epsilon =0$, reveals that this situation should correspond to a solution of (119, 120) with $u_t={\widetilde{u}}$, independently of $t$. This ansatz is indeed consistent with Eq. (119), and with this substitution Eq. (120) becomes

$$\begin{aligned} v_{t+1} = 1+ S({\widetilde{u}},v_t) \ . \end{aligned}$$

(140)

This last equation is a simple recursion on the $v$’s, with the initial value $v_0=1$. For the boundary condition $u_T=v_T$, $u_{T+1}=v_{T-1}$ to be asymptotically (when $T\rightarrow \infty$) verified one has to impose the values of ${\widetilde{u}}$ and $\lambda$ such that the $v_t$ solution of (140) converge to ${\widetilde{u}}$ when $t \rightarrow \infty$, in other words that the smallest fixed point solution $v \ge 1$ of $v=1+S({\widetilde{u}},v)$ is precisely equal to ${\widetilde{u}}$. The condition ${\widetilde{u}}=1+S({\widetilde{u}},{\widetilde{u}})$ imposes the following relationship between ${\widetilde{u}}$ and $\lambda$, ${\widetilde{u}}=1+\lambda {\widetilde{u}}^k$. Using this condition one can then rewrite (140) as

$$\begin{aligned} \frac{v_{t+1}}{{\widetilde{u}}} = \frac{1}{{\widetilde{u}}} + \left( 1-\frac{1}{{\widetilde{u}}}\right) \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \left( \frac{v_t}{{\widetilde{u}}}\right) ^p \left( 1-\frac{v_t}{{\widetilde{u}}}\right) ^{k-p} \ . \end{aligned}$$

(141)

Comparing this equation with (3) one realizes that by definition of ${\theta _\mathrm{r}}$, all the values of ${\widetilde{u}}$ in the interval $[1,1/{\theta _\mathrm{r}}[$ are such that the condition $v_t \rightarrow {\widetilde{u}}$ is fulfilled (with the value of $\lambda$ fixed by ${\widetilde{u}}=1+\lambda {\widetilde{u}}^k$). Let us now compute the RS thermodynamic quantities associated with this solution. As the $u_t$ are independent of $t$ the summation in Eq. (124) can be performed with a telescopic identity, and yields after a short computation ${F_\mathrm{site}}={\widetilde{u}}-1$. Similarly one sees easily from (125) that ${F_\mathrm{edge}}={\widetilde{u}}$ for this solution. This gives indeed the function $s(\theta )=-\theta \ln \theta - (1-\theta ) \ln (1-\theta )$ for $\theta > {\theta _\mathrm{r}}$ upon replacing in the expression of the RS thermodynamic potential (cf. Eq. (126)). In addition the cumulative distribution $P_t$ of activation times defined in Eq. (127) coincides on this solution with the series $x_t$ of Eq. (2) obtained as the activation time cumulative distribution of a random initial condition.

In the following we shall describe the non-trivial part of the resolution of the RS and energetic 1RSB equations in the large $T$ limit, i.e. in the RS case the part of the curve $s(\theta )$ for $\theta < {\theta _\mathrm{r}}$. The cases $l=k$ and $l<k$ are technically rather different, we shall thus divide the discussion according to this distinction.

1.2.2 Asymptotics for $l=k$

As explained in Sect. 1 in the case $l=k$ the equations on $v_t$ decouple, these quantities become independent of $T$ and are solutions of the recurrence $v_{t+1}=1+\lambda v_t^k$. A straightforward study of this equation (see Fig. 14 for an illustration) reveals the existence of a critical value $\lambda _\mathrm{c}$ such that $v_t$ converges to a finite value when $t\rightarrow \infty$ if $\lambda \le \lambda _\mathrm{c}$, while it diverges when $\lambda > \lambda _\mathrm{c}$. This critical parameter and the associated fixed-point $v_\mathrm{c}$ of the recurrence are solution of the equations:

$$\begin{aligned} v_\mathrm{c} = 1+ \lambda _\mathrm{c} v_c^k \ , \qquad 1 = \lambda _\mathrm{c} k \, v_\mathrm{c}^{k-1} \ , \end{aligned}$$

(142)

which are easily solved and yield $\lambda _\mathrm{c}=\frac{(k-1)^{k-1}}{k^k}$, $v_\mathrm{c}=\frac{k}{k-1}$.

Fig. 14

A graphical representation of the recursion $v_{t+1}=1+\lambda v_t^k$ (here for $k=2$). The dashed straight line corresponds to $v_{t+1}=v_t$, the three solid curves are, from bottom to top, for $\lambda < \lambda _\mathrm{c}$, $\lambda = \lambda _\mathrm{c}$ and $\lambda > \lambda _\mathrm{c}$

The case $\lambda < \lambda _\mathrm{c}$ corresponds actually to the trivial solution already discussed above, let us thus consider the alternative situation, $\lambda > \lambda _\mathrm{c}$. The divergence of $v_t$ is then actually very steep, with a double exponential form. Indeed when $v_t \gg 1$ the recurrence becomes approximately $v_{t+1} \approx \lambda v_t^k$, which reveals that $(\ln \ln v_t)/t$ converges to $\ln k$. As $u_0 \ge v_T$ one also has a divergence of $u_0$ with $T$ in this regime; from (126) (resp. (132)) this implies that the chemical potential $\mu$ of the RS formalism (resp. the parameter $y$ of the energetic 1RSB one) go to $-\infty$ (resp. $+\infty$), i.e. that the parametric curve $s(\theta )$ (resp. $\Sigma _\mathrm{e}(\theta )$) has a vertical tangent in this regime. Furthermore we shall prove now that the corresponding density $\theta$ of initially active sites converges to $(k-1)/(2k)$ (both in the RS and energetic 1RSB cases), hence this branch corresponds to a vertical segment. This is actually a consequence of the following statement on the behaviour of the functions ${F_\mathrm{site}}$ and ${F_\mathrm{edge}}$ of Eqs. (138, 125):

$$\begin{aligned} \forall \lambda > \lambda _\mathrm{c} \ , \quad \lim _{T\rightarrow \infty } {F_\mathrm{site}}(\lambda ,T) = \frac{k+1}{k-1} \ , \qquad \lim _{T\rightarrow \infty } {F_\mathrm{edge}}(\lambda ,T) = \frac{2k}{k-1} \ , \end{aligned}$$

(143)

as can be easily deduced from the expressions of $\theta$ given in (123, 126) and (133), along with the divergence of $u_0$ in the latter case. To prove the claim of Eq. (143), let us first note that, iterating (137), one obtains

$$\begin{aligned} u_t-u_{t+1}&= (u_0-u_1) \frac{1}{k^t} \frac{1}{(\lambda v_0^{k-1}) (\lambda v_1^{k-1}) \ldots (\lambda v_{t-1}^{k-1})} \end{aligned}$$

(144)

$$\begin{aligned}&= (u_0-u_1) \frac{1}{k^t} \frac{v_1 v_2 \ldots v_{t-1}}{(v_1-1)(v_2-1) \ldots (v_t -1)} \ , \end{aligned}$$

(145)

where we used (135) to go from the first to the second line. We can thus write

$$\begin{aligned} u_t-u_{t+1} = (u_0-u_1) \frac{1}{k^t} \alpha _t \frac{1}{v_t} \ , \end{aligned}$$

(146)

where we introduced the sequence $\alpha _t$ (note its independence on $T$) as

$$\begin{aligned} \alpha _t = \prod _{t^{\prime }=1}^t \frac{v_{t^{\prime }}}{v_{t^{\prime }}-1} \ , \qquad \alpha _0=1. \end{aligned}$$

(147)

We also have, in terms of this series,

$$\begin{aligned} u_0-u_1 = k^T \frac{1}{\alpha _T} v_T (v_T - v_{T-1}) \ . \end{aligned}$$

(148)

Using these relations, along with the representation $u_0 = v_T + \sum _{t=0}^{T-1} (u_t -u_{t+1})$, allows to rewrite the definition of (125) as:

$$\begin{aligned} {F_\mathrm{edge}}= \frac{ \alpha _T \frac{1}{k^T} \frac{v_T}{v_T-v_{T-1}} + 2 \sum \limits _{t=0}^{T-1} \alpha _t \frac{1}{k^t} }{\alpha _T \frac{1}{k^T} \frac{1}{v_T-v_{T-1}} + \sum \limits _{t=0}^{T-1} \frac{\alpha _t}{v_t} \frac{1}{k^t}} \ . \end{aligned}$$

(149)

The sum in the denominator can be transformed by noting that, from the definition of $\alpha _t$, $\alpha _t /v_t = \alpha _t - \alpha _{t-1}$. This yields

$$\begin{aligned} {F_\mathrm{edge}}= \frac{ \alpha _T \frac{1}{k^T} \frac{v_T}{v_T-v_{T-1}} + 2 \sum \limits _{t=0}^{T-1} \alpha _t \frac{1}{k^t} }{\alpha _T \frac{1}{k^T} \frac{1}{v_T-v_{T-1}} + \frac{1}{k^T} \alpha _{T-1} + \frac{k-1}{k}\sum \limits _{t=0}^{T-1} \alpha _t \frac{1}{k^t}} \ . \end{aligned}$$

(150)

Notice now that $\alpha _t$ has a finite limit when $t \rightarrow \infty$, thanks to the divergence of $v_t$ (for the limit of $\alpha _t$ to exists it is actually enough that $v_t \gg t$). Hence the summations in the above equation converge when $T\rightarrow \infty$ thanks to the exponentially decaying factor $1/k^t$, and all other terms in the numerator and denominator are neglectible in this limit. This proves the limit $2k/(k-1)$ for ${F_\mathrm{edge}}$ (one could also compute the main correction, of order $k^{-T}$, from this expression). The statement on ${F_\mathrm{site}}$ is proved with similar manipulations, that brings from (138) to the expression (exact for all $T$),

$$\begin{aligned} {F_\mathrm{site}}= \frac{ \alpha _T \frac{1}{k^T} \frac{v_{T-1}(v_T-1)}{v_T (v_T-v_{T-1})} + \frac{k+1}{k} \sum \limits _{t=0}^{T-1} \alpha _t \frac{1}{k^t} }{\alpha _T \frac{1}{k^T} \frac{1}{v_T-v_{T-1}} + \frac{1}{k^T} \alpha _{T-1} + \frac{k-1}{k}\sum \limits _{t=0}^{T-1} \alpha _t \frac{1}{k^t}} \ . \end{aligned}$$

(151)

As above the limit $T\rightarrow \infty$ can now be taken safely, the converging summations being the only non-vanishing terms of the numerator and denominator, hence the convergence of ${F_\mathrm{site}}$ to $(k+1)/(k-1)$, with corrections of order $k^{-T}$. These corrections actually contribute to the non-trivial dependence on $\lambda$ of $s$ and $\Sigma _\mathrm{e}$ (which are both finite) in this regime; we did not push their determination further, and merely observe here that their order $k^{-T}$ explains the statement on the finite $T$ corrections to ${\theta _\mathrm{min}}$ for $k=l=2$ and $k=l=3$ made in Sect. 4.2.1.

We have just seen that in the $T \rightarrow \infty$ limit the cases $\lambda < \lambda _\mathrm{c}$ and $\lambda > \lambda _\mathrm{c}$ describe, respectively, the trivial branch $\theta > {\theta _\mathrm{r}}$ of the RS entropy and its vertical segment at ${\theta _\mathrm{r}}/2$. To describe the range $[{\theta _\mathrm{r}}/2,{\theta _\mathrm{r}}]$ of non-trivial densities of initially active sites one has thus to investigate a regime where $\lambda$ is in a $T$-dependent scaling window around $\lambda _\mathrm{c}$.

Let us denote ${\widetilde{v}}_t$ the solution of the recursion right at the critical point, i.e. ${\widetilde{v}}_{t+1}= 1 + \lambda _\mathrm{c}{\widetilde{v}}_t^k$, with ${\widetilde{v}}_0=1$. This series converges to $v_\mathrm{c}$, with an asymptotic behaviour which is easily found to be

$$\begin{aligned} {\widetilde{v}}_t = v_\mathrm{c} - \frac{2k}{(k-1)^2} \frac{1}{t} + O\left( \frac{1}{t^2} \right) \ . \end{aligned}$$

(152)

Now if $\lambda = \lambda _\mathrm{c} + \delta$, with an infinitesimal positive value of $\delta$, the solution $v_t$ of the recursion $v_{t+1}=1+\lambda v_t^k$ spends a time of order $\delta ^{-1/2}$ around the avoided fixed-point $v_\mathrm{c}$ before crossing over to the doubly exponentially growing regime investigated above (this is a general feature of such recursive equations in the neighbourhood of a bifurcation, see for instance [22]). It is thus natural to investigate the scaling window parametrized by ${\widehat{\lambda }}$ as

$$\begin{aligned} \lambda = \lambda _\mathrm{c} + 2\pi ^2 \frac{(k-1)^{k-2}}{k^{k-1}}\frac{{\widehat{\lambda }}^2}{T^2} \ , \end{aligned}$$

(153)

the numerical prefactor and the square on ${\widehat{\lambda }}$ being chosen to simplify the following expressions. One can then look for a solution of the recurrence equation under the form $v_t = v_\mathrm{c} + \frac{1}{T} V(t/T)$, with $V(s)$ a scaling function. Expanding at the leading order in $T$ one obtains a differential equation on $V$,

$$\begin{aligned} V^{\prime }(s)=\frac{2\pi ^2 k {\widehat{\lambda }}^2}{(k-1)^2} + \frac{(k-1)^2}{2 k} V(s)^2 \ . \end{aligned}$$

(154)

The latter can be integrated into

$$\begin{aligned} V(s) = - \frac{2 k}{(k-1)^2} \frac{\pi {\widehat{\lambda }}}{\tan (\pi {\widehat{\lambda }}s)} \ , \end{aligned}$$

(155)

the constant in the solution of the differential equation being obtained by a matching argument between the regime $s\rightarrow 0$ and the large $t$ asymptotics of the critical series ${\widetilde{v}}_t$ given in (152). Note that this form is only valid for ${\widehat{\lambda }}< 1$, otherwise one enters the regime where $v_T$ diverges with $T$. One can furthermore assume a similar scaling ansatz for the $u_t$, introducing a scaling function $U(s)$ under the form $u_t = v_\mathrm{c} + U(t/T)$. Inserting these forms in Eq. (137) yields a differential equation on $U$,

$$\begin{aligned} \frac{U^{\prime \prime }(s)}{U^{\prime }(s)}=-\frac{(k-1)^2}{k} \, V(s) \ , \end{aligned}$$

(156)

which is integrated in

$$\begin{aligned} U^{\prime }(s) = B \sin ^2(\pi {\widehat{\lambda }}s) \ , \qquad U(s)=A + \frac{B}{2} \left( s - \frac{\sin (2 \pi {\widehat{\lambda }}s)}{2 \pi {\widehat{\lambda }}} \right) \ , \end{aligned}$$

(157)

with $A$ and $B$ two constants of integration. These can be fixed by imposing the boundary conditions $u_T=v_T$ and $u_{T+1}=v_{T-1}$, which translates here in $U(1)=V(1)/T$ and $U^{\prime }(1)=-V^{\prime }(1)/T$. Solving these equations yield $A$ and $B$; considering in particular $u_0=v_\mathrm{c} + U(0)$ one obtains, at the leading order in a large $T$ expansion,

$$\begin{aligned} u_0 = v_\mathrm{c} + \frac{1}{T} \frac{{\widehat{\lambda }}^2}{\sin ^4(\pi {\widehat{\lambda }})} \left( 1-\frac{\sin (2\pi {\widehat{\lambda }})}{2 \pi {\widehat{\lambda }}} \right) \frac{\pi ^2 k}{(k-1)^2} - \frac{1}{T} \frac{{\widehat{\lambda }}}{\tan (\pi {\widehat{\lambda }})} \frac{2 \pi k}{(k-1)^2} \ . \end{aligned}$$

(158)

One realizes at this point that for any fixed ${\widehat{\lambda }}< 1$, the limit of $u_0$ coincides with $v_\mathrm{c}$, in other words we are describing in this regime the end of the trivial branch, with $\theta \approx {\theta _\mathrm{r}}$. To describe the non-trivial regime of densities $[{\theta _\mathrm{r}}/2,{\theta _\mathrm{r}}]$ one has thus to further refine the scaling window, taking now ${\widehat{\lambda }}$ approaching $1$ in a $T$-dependent way. The inspection of (158) reveals that the correct scaling that allows to obtain a non-trivial limit of $u_0$ corresponds to ${\widehat{\lambda }}= 1 - O(T^{-1/4})$. We shall thus set

$$\begin{aligned} {\widehat{\lambda }}= 1 - \frac{1}{\sqrt{\pi }} \left( \frac{{\widetilde{\lambda }}}{(k-1) T} \right) ^\frac{1}{4}\!, \end{aligned}$$

(159)

with ${\widetilde{\lambda }}>0$ the new parameter describing this scale, the numerical prefactor being chosen for convenience. After a short computation one obtains the limit as $T \rightarrow \infty$ of the thermodynamic quantities in this scaling regime of $\lambda$ as

$$\begin{aligned} u_0({\widetilde{\lambda }}) = \frac{k}{k-1}\, \frac{1+ {\widetilde{\lambda }}}{{\widetilde{\lambda }}} \ , \quad {F_\mathrm{site}}({\widetilde{\lambda }}) = \frac{1}{k-1}\, \frac{k+1+ {\widetilde{\lambda }}}{1 + {\widetilde{\lambda }}} \ , \quad {F_\mathrm{edge}}({\widetilde{\lambda }}) = \frac{k}{k-1}\, \frac{2 + {\widetilde{\lambda }}}{1 + {\widetilde{\lambda }}} \ , \end{aligned}$$

(160)

the last two expressions being obtained by inserting the scaling ansatz on $u_t$ and $v_t$ in the definitions (125, 138); at the lowest order one can actually replace the $v_t$’s by $v_\mathrm{c}$ there. This yields a parametric representation of the thermodynamic quantities of the RS (resp. energetic 1RSB) formalism in terms of ${\widetilde{\lambda }}$, by inserting these last results in Eq. (126) (resp. (132, 133)). In the RS case one can check that ${\widetilde{\lambda }}\rightarrow 0$ corresponds to $\theta \rightarrow {\theta _\mathrm{r}}/2$, while ${\widetilde{\lambda }}\rightarrow \infty$ yields $\theta \rightarrow {\theta _\mathrm{r}}$, hence this scaling regime allows to cover the desired range $[{\theta _\mathrm{r}}/2,{\theta _\mathrm{r}}]$ for the densities of initially active sites. It is furthermore possible to invert the relation $\theta ({\widetilde{\lambda }})$, which yields finally the formula (82) announced in the main text for the entropy of activating initial configurations of density in the non-trivial interval $[{\theta _\mathrm{r}}/2,{\theta _\mathrm{r}}]$. In the energetic 1RSB case this last step does not seem possible and the final result (84) is presented in a form parametrized by ${\widetilde{\lambda }}$. We did not embark in a systematic study of the finite $T$ corrections in this regime, it is however clear that they are polynomially small in $T$, which justifies the statement made in Sect. 4.2.1 on the corrections to ${\theta _\mathrm{min}}(T)$ for $k=l\ge 4$.

Let us finally justify the results presented at the end of Sect. 4.2.1 on the distribution of activation times. Assuming a finite value of $t$, the expression of (139) becomes in the regime parametrized by ${\widetilde{\lambda }}$:

$$\begin{aligned} {F_\mathrm{site}}({\widetilde{\lambda }},t) = \frac{\lambda _\mathrm{c}}{u_0({\widetilde{\lambda }})} \left[ {\widetilde{v}}_{t-1}^{k+1} + (k+1) {\widetilde{v}}_{t-1}^k (u_0({\widetilde{\lambda }}) - {\widetilde{v}}_{t-1}) \right] , \end{aligned}$$

(161)

the last summation in (139) yielding a subdominant correction of order $1/T$. Note that ${F_\mathrm{site}}({\widetilde{\lambda }},t)$ tends to ${F_\mathrm{site}}({\widetilde{\lambda }})$ as $t\rightarrow \infty$, which means that the support of the distribution of the activation times does not scale with $T$ in this regime. The expression (92) for the cumulative distribution of activation times follows then easily from its generic definition given in Eq. (127), upon expressing all the quantities depending on ${\widetilde{\lambda }}$ as a function of the corresponding $\theta$. In the main text we introduced for clarity the series $w_t = {\theta _\mathrm{r}}{\widetilde{v}}_t$, to allow for an easier comparison with the distribution of activation times from a random initial condition.

1.2.3 Asymptotics for $l<k$

Let us now discuss the solution of the set of Eqs. (119, 120) in the limit $T\rightarrow \infty$, in the case $l<k$, and justify the statements made in Sect. 4.2.2; as we shall see their behaviour and the method of study is qualitatively different compared to the case $l=k$.

We shall first rephrase Eqs. (119, 120) as a single recursive equation, by introducing a four-dimensional vector $w_t$ defined by

$$\begin{aligned} w_t=\begin{pmatrix} u_{t} \\ u_{t+1} \\ v_{t} \\ v_{t-1} \end{pmatrix}. \end{aligned}$$

(162)

The recursive equations (119, 120) on the $u_t$’s and $v_t$’s become a single recursion on $w_t$, of the form $w_{t+1} = R(w_t)$ where the function $R$ is given by

$$\begin{aligned} R \begin{pmatrix} u_{} \\ u_+ \\ v_{} \\ v_- \end{pmatrix} = \begin{pmatrix} u_+ \\ E(u,u_+,v) \\ v+S(u_+,v)-S(u_+,v_-) \\ v \end{pmatrix}. \end{aligned}$$

(163)

The function $S$ was defined in (121), while $E(u,u_+,v)$ is given implicitly as $D(E(u,u_+,v),v)=D(u_+,v)+u_+-u$, with the function $D$ of (121). Inverting this relation one obtains an explicit expression of $E$:

$$\begin{aligned} E(u,u_+,v) = v+\left( (u_+-v)^{k-l+1} + \frac{1}{\lambda \left( {\begin{array}{c}k\\ l-1\end{array}}\right) } \frac{u_+-u}{v^{l-1}} \right) ^\frac{1}{k-l+1}\!. \end{aligned}$$

(164)

We have thus a representation of the time evolution of $w$ as the flow of a discrete dynamical system in a four-dimensional space. The boundary conditions on the $u_t$’s and $v_t$’s translate into conditions on the allowed values of $w_0$ and $w_T$. The former must indeed lie in the two-dimensional manifold with $v=1$ and $v_-=0$, while the latter is restricted to the two-dimensional manifold defined by $u=v$ and $u_+=v_-$. When $T\rightarrow \infty$, for a fixed value of $\lambda$, the solution $w_t$ of the recursion $w_{t+1}=R(w_t)$ must find a way to go infinitely slowly from the first manifold at $t=0$ to the second one at $t=T\rightarrow \infty$. It must in consequence remains as close as possible to the fixed points of the evolution map $R$.

The study of the equation $w=R(w)$ is very simple and shows that these fixed points span the two-dimensional subspace with $u=u_+$, $v=v_-$. One can then compute the Jacobian matrix of $R$ on such a fixed-point, and realizes that this matrix has two eigenvalues equal to 1 (corresponding to the invariance of the fixed-point subspace under $u\rightarrow u+\delta u$ and $v \rightarrow v+\delta v$), and two eigenvalues $C(u,v)$ and $1/C(u,v)$, where $C$ is the function defined in (122). All the fixed points have thus an unstable direction, except the one-dimensional set of fixed points obeying the further condition $C(u,v)=1$, which constitutes a line of marginal fixed points. In the $T\rightarrow \infty$ limit the solution $w_t$ is thus expected to remain close to this line, otherwise the flow along the unstable directions forbid to go from one boundary manifold at $t=0$ to the other one at $t=T \gg 1$. This analysis is corroborated by the numerical results presented in Fig. 15, where we show the solution $u_t,v_t$ determined numerically for some large but finite value of $T$. In particular the right panel demonstrate that for most values of $t$ (i.e. excluding both $t$ finite and $T-t$ finite in the large $T$ limit), the couple $(u_t,v_t)$ falls on the marginal fixed-point line $C(u,v)=1$.

Fig. 15

The solution of the Eqs. (119, 120) for $k=3$, $l=2$, with $\lambda =0.005$ and $T=400$. Left panel the solid curves are $u_t$ (top) and $v_t$ (bottom) as functions of $t$; the dashed horizontal lines correspond, from top to bottom, to $u_*$, ${\widehat{u}}$, ${\widehat{v}}$ and $v_*$, solutions of (166, 167). Right panel parametric plot of the same data, with symbols instead of lines to appreciate the discreteness in $t$. Dashed line is the solution of the equation $C(u,v)=1$, almost superimposed with most of the points $(v_t,u_t)$. The arrows point to the beginning $(v_*,u_*)$ and end $({\widehat{v}},{\widehat{u}})$ of the scaling regime along the curve $C(u,v)=1$

More precisely, the solution $u_t,v_t$ can be described in the large $T$ limit by two scaling functions $U(s)$ and $V(s)$, function of a rescaled time $s=t/T \in ]0,1[$, such that at the leading order,

$$\begin{aligned} u_t = U\left( \frac{t}{T}\right) \ , \qquad v_t = V\left( \frac{t}{T}\right) \ . \end{aligned}$$

(165)

Inserting this ansatz in the Eqs. (119, 120), one realizes that the condition $C(U(s),V(s))=1$, that we obtained intuitively above, is indeed precisely what is needed to enforce (119, 120) at the leading order in the large $T$ limit. Note that the explicit dependency of $U$ and $V$ on $s$ can be determined from the sub-dominant corrections in this limit; however we shall not need it in what follows. It will indeed be enough to compute the value of $U$ and $V$ for $t$ small and $t$ close to $T$, i.e. for $s$ around 0 and 1. As revealed by the numerical data presented in Fig. 15, the matching between the scaling regime described by the functions $U,V$ (i.e. for $s$ strictly between 0 and 1) and the boundary conditions at $t=0$ and $t=T$ affects the series $v_t$ but not $u_t$. In other words, for $t$ finite while $T\rightarrow \infty$ one has $u_t\rightarrow u_*=U(0)$ independently of $t$, where $u_*$ is some ($\lambda$ dependent) constant still to be determined, while $v_t$ converges to the solution of the recursion $v_{t+1}=v_t+S(u_*,v_t)-S(u_*,v_{t-1})$ obtained from (120) by replacing $u_t$ by its limit $u_*$. Equivalently one has in this regime $v_{t+1}=1+S(u_*,v_t)$. When $t\rightarrow \infty$ (after the large $T$ limit) this series $v_t$ converges to $v_*=V(0)$, the smallest fixed-point solution of this recursion on $v$; for this behaviour to match the beginning of the scaling regime (i.e. $s\rightarrow 0$) one must impose simultaneously

$$\begin{aligned} C(u_*,v_*)= 1 \ , \qquad \text {and} \ \ \ v_*=1+S(u_*,v_*) \ . \end{aligned}$$

(166)

The first equation allows to express $u_*$ as a function of $v_*$; replacing in the second one leads to the single equation on $v_*$ given in Eq. (96), while (97) is nothing but an explicit version of the condition $C(u_*,v_*)= 1$. A similar reasoning in the regime $T-t$ finite reveals that $U(1)={\widehat{u}}$ and $V(1)={\widehat{v}}$ have to obey

$$\begin{aligned} C({\widehat{u}},{\widehat{v}}) =1 \ , \qquad \text {and} \ \ \ {\widehat{v}}= S({\widehat{u}},{\widehat{v}}) + {\widehat{u}}- S({\widehat{u}},{\widehat{u}}) \ . \end{aligned}$$

(167)

It is easy to check that the expressions of ${\widehat{u}}$ and ${\widehat{v}}$ given in (95) are indeed solutions of these two equations, using the equations on ${\theta _\mathrm{r}}$ and ${\widetilde{x}_\mathrm{r}}$ of Eq. (4). By definition for $\lambda \in ]0,\lambda _\mathrm{r}]$ one has $u_* \ge {\widehat{u}}\ge {\widehat{v}}\ge v_*$, see Fig. 16 for a representation of the solution of the Eqs. (166, 167) as a function of $\lambda$. In $\lambda _\mathrm{r}$, where one recovers the trivial solution studied in Appendix section “The Trivial Solution”, one has $u_*={\widehat{u}}=1/{\theta _\mathrm{r}}$ and $v_*={\widehat{v}}={\widetilde{x}_\mathrm{r}}/{\theta _\mathrm{r}}$.

Fig. 16

The functions $u_*$, ${\widehat{u}}$, ${\widehat{v}}$ and $v_*$ (from top to bottom) solutions of Eqs. (166, 167) as a function of $\lambda$ for $k=3$, $l=2$. The upper two and lower two curves meet in $\lambda =\lambda _\mathrm{r}$. When $\lambda \rightarrow 0$ the upper three curves diverge, while $v_*$ converges to $l/(l-1)$

Let us now deduce the value of ${F_\mathrm{site}}$ and ${F_\mathrm{edge}}$ in the large $T$ limit from the above characterization of the behaviour of the $u_t$’s and $v_t$’s. From Eq. (125) one has in this limit

$$\begin{aligned} \lim _{T \rightarrow \infty }{F_\mathrm{edge}}(\lambda ,T) = \frac{1}{u_*}\left[ {\widehat{u}}^2 - 2 \int _0^1 \mathrm{d}s \, U^{\prime }(s) V(s) \right] \ , \end{aligned}$$

(168)

the matching regimes of $t$ finite and $T-t$ finite having neglectible contributions to the summation. The integral above can be computed even if we have not determined the time-dependency of the scaling functions $U(s)$ and $V(s)$: using $\mathrm{d}s \, U^{\prime }(s) = \mathrm{d}u$ and the condition $C(U(s),V(s))=1$, one has

$$\begin{aligned} -\int _0^1 \mathrm{d}s \, U^{\prime }(s) V(s) = \int _{{\widehat{u}}}^{u_*} \mathrm{d}u \ v(u) = u_* v_* - {\widehat{u}}\, {\widehat{v}}+ \int _{v_*}^{{\widehat{v}}} \mathrm{d}v \ u(v) \ , \end{aligned}$$

(169)

where $u(v)$ (resp. $v(u)$) is the solution of $C(u(v),v)=1$ (resp. $C(u,v(u))=1$). The equation $C(u(v),v)=1$ can be explicitly solved into

$$\begin{aligned} u(v) = v + \left( \lambda l \left( {\begin{array}{c}k\\ l\end{array}}\right) \right) ^{-\frac{1}{k-l}} v^{- \frac{l-1}{k-l}} \ . \end{aligned}$$

(170)

This allows to compute the integral in (169) and to obtain (99).

We shall now compute similarly the limit of ${F_\mathrm{site}}$ that was defined in Eq. (124). In that equation we shall exploit the fact that $u_t-u_{t+1}$ is of order $1/T$ to perform the approximation

$$\begin{aligned} (u_t-v_{t-2})^{k+1-p} = (u_{t-1}-v_{t-2})^{k+1-p}+(k+1-p) \nonumber \\ (u_t-u_{t-1})(u_{t-1}-v_{t-2})^{k-p} + O\left( \frac{1}{T^2} \right) \ . \end{aligned}$$

(171)

Within this approximation the first term leads to a telescopic summation, we then get

$$\begin{aligned}&{F_\mathrm{site}}\sim \frac{\lambda }{u_0} \sum _{p=l}^{k+1} \left( {\begin{array}{c}k+1\\ p\end{array}}\right) \nonumber \\&\quad \times \left[ v_{T-1}^p (u_T \!-\! v_{T-1})^{k+1-p} \!-\! (k\!+\!1-p) \sum _{t=1}^T v_{t-2}^p (u_t-u_{t-1})(u_{t-1}-v_{t-2})^{k-p} \right] \end{aligned}$$

(172)

As $u_T=v_{T-1}+O(1/T)$ in the first summation only the term $p=k+1$ survives; the second term can be rearranged as above in terms of integrals of the scaling functions, namely

$$\begin{aligned} \lim _{T\rightarrow \infty } {F_\mathrm{site}}(\lambda ,T)&= \frac{\lambda }{u_*}\left[ {\widehat{u}}^{k+1} - (k+1) \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \int _0^1 \mathrm{d}s \ U^{\prime }(s) V(s)^p (U(s)-V(s))^{k-p} \right] \end{aligned}$$

(173)

$$\begin{aligned}&= \frac{\lambda }{u_*}\left[ {\widehat{u}}^{k+1} + (k+1) \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \int _{{\widehat{u}}}^{u_*} \mathrm{d}u \ v(u)^p (u-v(u))^{k-p} \right] \end{aligned}$$

(174)

$$\begin{aligned}&= \frac{\lambda }{u_*}\left[ {\widehat{u}}^{k+1} + (k+1) \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \int _{v_*}^{{\widehat{v}}} \mathrm{d}v \ (-u^{\prime }(v)) v^p (u(v)-v)^{k-p} \right] \end{aligned}$$

(175)

Inserting the expression of $u(v)$ given in Eq. (170) yields easily to the value of ${F_\mathrm{site}}$ written in (98). The parametric representations of $s(\theta )$ and $\Sigma _\mathrm{e}(\theta )$ given in Sect. 4.2.2 are then direct consequences of Eqs. (126, 132, 133).

For what concerns the distribution of activation times, one has in the regime $t=sT$ with $s\in ]0,1[$ the following limit for the function ${F_\mathrm{site}}$ defined in (128):

$$\begin{aligned} \lim _{T\rightarrow \infty } {F_\mathrm{site}}(\lambda ,T,t=sT)\!&= \!\frac{\lambda }{u_*} \left[ \sum _{p=l}^{k+1} \left( {\begin{array}{c}k+1\\ p\end{array}}\right) V(s)^p (U(s) - V(s))^{k+1-p} \right. \nonumber \\&\left. -\, (k+1) \sum _{p=l}^k \left( {\begin{array}{c}k\\ p\end{array}}\right) \int _0^s \mathrm{d}s^{\prime } \ U^{\prime }(s^{\prime }) V(s^{\prime })^p (U(s^{\prime })-V(s^{\prime }))^{k-p} \right] \!.\nonumber \\ \end{aligned}$$

(176)

Studying the limit $s\rightarrow 0+$ and $s\rightarrow 1^-$ of this expression leads to the expressions (104) for the fraction of vertices which activate at the very beginning and at the very end of the process.