Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically constrained East model

Abstract

The East model is the simplest one-dimensional kinetically constrained model of N spins with a trivial equilibrium that displays anomalously large spatio-temporal fluctuations, with characteristic “space-time bubbles” in trajectory space, and with a discontinuity at the origin for the first derivative of the scaled cumulant generating function of the total activity. These striking dynamical properties are revisited via the large deviations at various levels for the relevant local empirical densities and activities that only involve two consecutive spins. This framework allows to characterize their anomalous rate functions and to analyze the consequences for all the time-additive observables that can be reconstructed from them, both for the pure and for the random East model. These singularities in dynamical large deviations properties disappear when the hard constraint of the East model is replaced by the soft constraint.

Appendices

Appendix A: Reminder on large deviations for the relevant empirical observables

In this Appendix, we summarize the general procedure to derive the large deviations properties for the relevant empirical observables of Markov trajectories.

1.1 A.1 Identification of the relevant time-empirical observables that determine the trajectories probabilities

For the Markov model defined by the Markov generator W, the first step consists in rewriting the probability of a long dynamical trajectory $C(0 \le t \le T)$

$$\begin{aligned} \mathcal{P}[C(0 \le t \le T)] \mathop {\simeq }_{T \rightarrow +\infty } e^{\displaystyle -T \Phi _{[W]} \left( E [C(0 \le t \le T)] \right) } \end{aligned}$$

(A1)

in terms of an intensive action $\Phi _{[W]} \left( E [C(0 \le t \le T)] \right)$ that depends on the Markov generator W, and that only involves a few relevant time-empirical observables $E [C(0 \le t \le T)]$ of the dynamical trajectory $C(0 \le t \le T)$.

1.2 A.2 Number of dynamical trajectories of length T with the same value of the time-empirical observables

Since all the individual dynamical trajectories $C(0 \le t \le T)$ that have the same empirical observables $E=E [C(0 \le t \le T)]$ have the same probability given by Eq. (A1), one can rewrite the normalization over all possible trajectories as a sum over these empirical observables

$$\begin{aligned} 1= \sum _{C(0 \le t \le T)} \mathcal{P}[C(0 \le t \le T)] \mathop {\simeq }_{T \rightarrow +\infty } \sum _{E} \Omega _T(E ) e^{\displaystyle -T \Phi _{[W]} \left( E \right) }, \nonumber \\ \end{aligned}$$

(A2)

where the number of dynamical trajectories of length T associated with the given values E of these empirical observables

$$\begin{aligned} \Omega _T ( E ) \equiv \sum _{C(0 \le t \le T)} \delta \left( E [C(0 \le t \le T)] - E \right) \end{aligned}$$

(A3)

grows exponentially with respect to the length T of the trajectories

$$\begin{aligned} \Omega _T( E) \mathop {\simeq }_{T \rightarrow +\infty } C(E) \ e^{\displaystyle T S( E ) }. \end{aligned}$$

(A4)

The prefactor C(E) denotes the appropriate constitutive constraints for the empirical observables E. The factor $S( E ) = \frac{\ln \Omega _T(E) }{ T }$ represents the Boltzmann intensive entropy of the set of trajectories of length T with given empirical observables E. Let us now recall how it can be evaluated without any actual computation (i.e., one does not need to use combinatorial methods to count the appropriate configurations).

The normalization of Eq. (A2) becomes for large T

$$\begin{aligned} 1 \mathop {\simeq }_{T \rightarrow +\infty } \sum _{E} C(E) \ e^{\displaystyle T \left[ S( E ) - \Phi _{[W]} \left( E \right) \right] }. \end{aligned}$$

(A5)

When the empirical variables E take their typical values $E_{[W]}^\mathrm{{typ}}$ for the Markov generator W, the exponential behavior in T of Eq. (A5) should exactly vanish, i.e., the entropy $S( E_{[W]}^\mathrm{{typ}})$ should exactly compensate the action $\Phi _{[W]} \left( E_{[W]}^\mathrm{{typ}}\right)$

$$\begin{aligned} S( E_{[W]}^\mathrm{{typ}} )= \Phi _{[W]} \left( E_{[W]}^\mathrm{{typ}}\right) . \end{aligned}$$

(A6)

To obtain the intensive entropy S(E) for any other given value E of the empirical observables, one just needs to introduce the modified Markov generator $\hat{W}_E$ that would make the empirical values E typical for this modified model

$$\begin{aligned} E = E_{[\hat{W}_E]}^\mathrm{{typ}} \end{aligned}$$

(A7)

and to use Eq. (A6) for this modified model to obtain

$$\begin{aligned} S( E) = S( E_{[\hat{W} (E)]}^\mathrm{{typ}} )= \Phi _{[{\hat{W}}_E]} \left( E_{\hat{W}_E}^\mathrm{{typ}} \right) = \Phi _{[\hat{W}_E]} \left( E \right) . \nonumber \\ \end{aligned}$$

(A8)

Here, one should stress the modified generator ${\hat{W}}_E$, and thus, S(E) depends only on the empirical observables E and do not involve the initial generator W. Plugging Eq. (A8) into Eq. (A4) yields that the number $\Omega _T ( E )$ of dynamical trajectories of length T associated with given values E of these empirical observables of Eq. (A3)

$$\begin{aligned} \Omega _T ( E ) \mathop {\simeq }_{T \rightarrow +\infty } C(E) \ e^{\displaystyle T S( E ) } \mathop {\simeq }_{T \rightarrow +\infty } C(E) \ e^{\displaystyle T \Phi _{[\hat{W}_E]} \left( E \right) } \nonumber \\ \end{aligned}$$

(A9)

only involves the action $\Phi _{[\hat{W}_E]} \left( E \right)$ of the empirical observables E evaluated for the modified generator ${\hat{W}}_E$ defined by Eq. (A7).

1.3 A.3 Large deviations for the relevant time-empirical observables E

The normalization over trajectories of Eq. (A2) can be rewritten as the normalization

$$\begin{aligned} 1 = \sum _{ E } P_T^{[2.5]} (E) \end{aligned}$$

(A10)

for the probability

$$\begin{aligned} P_T^{[2.5]} (E) \mathop {\simeq }_{T \rightarrow +\infty } \Omega _T(E ) e^{\displaystyle -T \Phi _{[W]} (E) } \end{aligned}$$

(A11)

to see the empirical observables E when the dynamical trajectories of length T are governed by the Markov generator W. Plugging Eq. (A9) into Eq. (A2) yields the large deviation form

$$\begin{aligned} P^{[2.5]}_T (E) \mathop {\simeq }_{T \rightarrow +\infty } C(E) \ e^{\displaystyle - T I_{2.5} (E) } \end{aligned}$$

(A12)

where the rate function at Level 2.5

$$\begin{aligned} I_{2.5} (E) = \Phi _{[W]} (E) - \Phi _{[\hat{W}_E]} (E) \end{aligned}$$

(A13)

is simply given by the difference between the intensive action $\Phi _{[W]} (E)$ associated with the true generator W and the intensive action $\Phi _{[\hat{W}_E]} (E)$ associated with the modified generator $\hat{W}_E$ that would make the empirical value E typical (see Eq. A7). It is positive $I_{2.5} (E) \ge 0$ and vanishes when E takes the typical value $E_{[W]}^\mathrm{{typ}}$

$$\begin{aligned} 0=I_{2.5} (E_{[W]}^\mathrm{{typ}}), \end{aligned}$$

(A14)

i.e., only when the modified generator $\hat{W}_E$ coincides with the true generator W.

1.4 A.4 Example : derivation of the large deviations at Level 2.5 for Markov jump processes

Let us now describe how the general formalism described above can be applied to the continuous-time Markov jump process described by the Master equation

$$\begin{aligned} \frac{\partial P_t(C)}{\partial t} = \sum _{C' } W(C,C') P_t(C'). \end{aligned}$$

(A15)

The trajectory probability of Eq. (10) can be rewritten as

$$\begin{aligned}&\mathcal{P}[x(0 \le t \le T)] \nonumber \\&\quad = e^{ \displaystyle T \sum _{C }\sum _{ C' \ne C } \left[ q(C',C) \ln ( W(C',C) ) - \rho (C) W(C',C) \right] } \end{aligned}$$

(A16)

in terms of the empirical time-averaged density

$$\begin{aligned} \rho (C) \equiv \frac{1}{T} \int _0^T \mathrm{{d}}t \ \delta _{C(t),C} \end{aligned}$$

(A17)

satisfying the normalization

$$\begin{aligned} \sum _C \rho (C) = 1 \end{aligned}$$

(A18)

and on the empirical flows from C to $C' \ne C$

$$\begin{aligned} q(C',C) \equiv \frac{1}{T} \sum _{t : C(t) \ne C(t^+)} \delta _{C(t^+),C'} \delta _{C(t),C} \end{aligned}$$

(A19)

satisfying the following stationarity constraints. For any configurations C, the total incoming flow into C

$$\begin{aligned} q_{in}(C) \equiv \sum _{C' \ne C} q(C,C') = \frac{1}{T} \sum _{t : C(t) \ne C(t^+)} \delta _{C(t^+),C} \nonumber \\ \end{aligned}$$

(A20)

and the total outgoing flow from C

$$\begin{aligned} q_{out}(C) \equiv \sum _{C' \ne C} q(C',C) = \frac{1}{T} \sum _{t : C(t) \ne C(t^+)} \delta _{C(t),C} \nonumber \\ \end{aligned}$$

(A21)

should be equal up to boundary terms of order 1/T (involving the initial configuration at time $t=0$ and the final configuration at time T) that can be neglected for large time-window $T \rightarrow +\infty$

$$\begin{aligned} 0=q_{out}(C) - q_{in}(C) = \sum _{C' \ne C} \left( q(C',C)- q(C,C') \right) . \nonumber \\ \end{aligned}$$

(A22)

With respect to the general formalism summarized in Appendix 1, this means that the relevant empirical observables E are the empirical density $\rho (.)$ and the empirical flows q(., .), while the corresponding action introduced in Eq. (A1) reads

$$\begin{aligned}&\Phi _{[W]} [ \rho (.) ; q(.,.)] = \sum _{C }\sum _{ C' \ne C }\nonumber \\&\quad \left[ \rho (C) W(C',C) - q(C',C) \ln ( W(C',C) ) \right] . \nonumber \\ \end{aligned}$$

(A23)

For the modified rates $\hat{W}(C',C)$ that would make typical the empirical variables $[ \rho (.) ; q(.,.)]$

$$\begin{aligned} \hat{W}(C',C) = \frac{ q(C',C) }{\rho (C) }, \end{aligned}$$

(A24)

the action of Eq. (A23) becomes

$$\begin{aligned}&\Phi _{[\hat{W}]} [ \rho (.) ; q(.,.)]\nonumber \\&\quad = \sum _{C }\sum _{ C' \ne C } \left[ \rho (C) \hat{W}(C',C) - q(C',C) \ln ( \hat{W}(C',C) ) \right] \nonumber \\&\quad = \sum _{C }\sum _{ C' \ne C } \left[ q(C',C) - q(C',C) \ln ( \frac{ q(C',C) }{\rho (C) } ) \right] . \end{aligned}$$

(A25)

The difference of Eq. (A13) between the actions of Eqs. (A23) and (A25) allows to recover the well-known rate function at Level 2.5

$$\begin{aligned} I_{2.5}( \rho _. ; q_{.,.} )&= \Phi _{[W]} [ \rho (.) ; q(.,.)] - \Phi _{[\hat{W}]} [ \rho (.) ; q(.,.)] \nonumber \\&= \sum _{C } \sum _{C' \ne C} \left[ q(C',C) \ln \left( \frac{ q(C',C) }{ W(C',C) \rho (C) } \right) \right. \nonumber \\&\quad \left. - q(C',C) + W(C',C) \rho (C) \right] \end{aligned}$$

(A26)

that governs the probability of the empirical observables $[ \rho (.) ; q(.,.)]$ for large T [21, 24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]

$$\begin{aligned}&P^{[2.5]}_{T}[ \rho (.) ; q(.,.) ] \mathop {\propto }_{T \rightarrow +\infty } e^{- T I_{2.5}[ \rho (.) ; q(.,.) ] } \delta \nonumber \\&\quad \left( \sum _C \rho (C) - 1 \right) \prod _C \delta \left[ \sum _{C' \ne C} (q(C,C') - q(C',C) ) \right] , \nonumber \\ \end{aligned}$$

(A27)

where the constitutive constraints of the empirical observables have been discussed in Eqs. (A18) and (A22).

Appendix B: Detailed-balance Markov jump processes: explicit contractions of the Level 2.5

In this Appendix, we describe how the Level 2.5 of Markov Jump processes described in Sect. 1 of the previous Appendix can be contracted explicitly towards lower levels when the rates satisfy the detailed-balance condition.

1.1 B.1 Level 2.5 when the empirical flows are replaced by the empirical activities and the empirical currents

It is convenient to order the configurations. For each link $(C'>C)$, it is useful to parametrize the two empirical flows $q(C',C)$ and $q(C,C')$ of Eq. (A19)

$$\begin{aligned} q(C',C) \equiv \frac{a(C',C) +j(C',C) }{2} \nonumber \\ q(C,C') \equiv \frac{a(C',C) -j(C',C) }{2} \end{aligned}$$

(B1)

by their symmetric and antisymmetric parts called the activity and the current

$$\begin{aligned} a(C',C) \equiv q(C',C) +q(C,C') = a(C,C') \nonumber \\ j(C',C) \equiv q(C',C) -q(C,C') = - j(C,C'). \end{aligned}$$

(B2)

Since the stationarity constraints of Eq. (A22) only involve the currents j(., .) and not the activities a(., .)

$$\begin{aligned} \sum _{C' \ne C} j(C',C) =0, \end{aligned}$$

(B3)

the Level 2.5 of Eq. (A27) becomes

$$\begin{aligned}&P^{[2.5]}_{T}[ \rho (.) ; a(.,.) ; j(.,.) ] \mathop {\propto }_{T \rightarrow +\infty } e^{- T I_{2.5}[ \rho (.) ; a(.,.) ; j(.,.) ] }\nonumber \\&\quad \delta \left( \sum _C \rho (C) - 1 \right) \prod _C \delta \left[ \sum _{C' \ne C} j(C',C) \right] . \end{aligned}$$

(B4)

The rate function translated from Eq. (A26) via the change of variables of Eq. (B1)

$$\begin{aligned}&I_{2.5}[ \rho (.) ; a(.,.) ; j(.,.) ]= \sum _{C } \sum _{C' > C}\nonumber \\&\quad I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ] \end{aligned}$$

(B5)

involves the following contribution for the link $[C',C]$:

$$\begin{aligned}&I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]\nonumber \\&\quad \equiv \frac{j(C',C) }{2} \ln \left( \frac{ (a(C',C) +j(C',C)) W(C,C') \rho (C') }{ (a(C',C) -j(C',C)) W(C',C) \rho (C) } \right) \nonumber \\&\qquad + \frac{a(C',C) }{2} \ln \left( \frac{ a^2(C',C) - j^2(C',C) }{ 4 W(C',C) \rho (C) W(C,C') \rho (C')} \right) \nonumber \\&\qquad - a(C',C) + W(C',C) \rho (C) + W(C,C') \rho (C'). \nonumber \\ \end{aligned}$$

(B6)

It is useful to separate the even and the odd parts with respect to the link current $j(C',C)$

$$\begin{aligned}&I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ] \nonumber \\&\quad = I^{[C',C]\mathrm{{Even}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]\nonumber \\&\qquad + I^{[C',C]\mathrm{{Odd}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]. \ \ \nonumber \\ \end{aligned}$$

(B7)

The even contribution reads

$$\begin{aligned}&I^{[C',C]\mathrm{{Even}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]\nonumber \\&\quad \equiv \frac{I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]+I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; -j(C',C) ]}{2}\nonumber \\&\quad = \frac{j(C',C) }{2} \ln \left( \frac{ a(C',C) +j(C',C)}{a(C',C) -j(C',C) } \right) + \frac{a(C',C) }{2} \ln \left( \frac{ a^2(C',C) - j^2(C',C) }{ 4 W(C',C) \rho (C) W(C,C') \rho (C')} \right) \nonumber \\&\qquad - a(C',C) + W(C',C) \rho (C) + W(C,C') \rho (C'), \end{aligned}$$

(B8)

while the odd contribution

$$\begin{aligned}&I^{[C',C]\mathrm{{Odd}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ] \nonumber \\&\quad \equiv \frac{I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]-I^{[C',C]}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; -j(C',C) ]}{2} \nonumber \\&\quad = \frac{j(C',C) }{2} \ln \left( \frac{ W(C,C') \rho (C') }{ W(C',C) \rho (C) } \right) \end{aligned}$$

(B9)

is simply linear in the current $j(C',C)$. The factor $\ln \left( \frac{ W(C,C') \rho (C') }{ W(C',C) \rho (C) } \right)$ measures the irreversibility associated with the two link flows $W(C,C') \rho (C')$ and $W(C',C) \rho (C)$ that would be typically produced by the empirical densities $\rho (C')$ and $\rho (C)$. This can be considered as an example of the Gallavotti–Cohen fluctuation relations (see [33, 48, 87,88,89,90,91,92,93,94,95,96,97,98] and references therein).

1.2 B.2 Detailed-balance: the sum of the current-odd contributions vanishes in the rate function at Level 2.5

When the rates satisfy the detailed-balance condition on each link $C \ne C'$

$$\begin{aligned} 0= W(C,C') P_\mathrm{{eq}} (C') - W(C',C) P_\mathrm{{eq}}(C) \end{aligned}$$

(B10)

on can plug the ratio of the two rates

$$\begin{aligned} \frac{ W(C,C') }{ W(C',C)} = \frac{ P_\mathrm{{eq}}(C) }{ P_\mathrm{{eq}} (C') } \end{aligned}$$

(B11)

into the odd contribution of Eq. (B9) to obtain

$$\begin{aligned}&I^{[C',C]\mathrm{{Odd}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]\nonumber \\&\quad = \frac{j(C',C) }{2} \left[ \ln \left( \frac{ \rho (C') }{ P_\mathrm{{eq}}(C') } \right) - \ln \left( \frac{ \rho (C) }{ P_\mathrm{{eq}}(C) } \right) \right] . \nonumber \\ \end{aligned}$$

(B12)

The sum of all the odd contributions of Eq. (B12) then vanishes as a consequences of the antisymmetry of the current $j(C',C) = - j(C,C')$ of Eq. (B2) and of the stationarity constraint of Eq. (B3)

$$\begin{aligned}&\sum _{C } \sum _{C'> C} I^{[C',C]\mathrm{{Odd}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ] \nonumber \\&\quad = \sum _{C } \sum _{C'> C} \frac{j(C',C) }{2} \ln \left( \frac{ \rho (C') }{ P_\mathrm{{eq}}(C') } \right) \nonumber \\&\qquad - \sum _{C } \sum _{C'> C} \frac{j(C',C) }{2} \ln \left( \frac{ \rho (C) }{ P_\mathrm{{eq}}(C) } \right) \nonumber \\&\quad = - \frac{1}{2} \sum _{C } \ln \left( \frac{ \rho (C) }{ P_\mathrm{{eq}}(C) } \right) \left[ \sum _{C'<C} j(C',C) + \sum _{C' > C} j(C',C) \right] \nonumber \\&\quad = - \frac{1}{2} \sum _{C } \ln \left( \frac{ \rho (C) }{ P_\mathrm{{eq}}(C) } \right) \left[ \sum _{C'\ne C} j(C',C) \right] =0. \end{aligned}$$

(B13)

As recalled after Eq. (B9), the odd contributions of the links to the rate function at Level 2.5 are linear in the link currents and are directly related to the irreversibility of the dynamics. The physical interpretation of the vanishing of the sum of all these odd contributions (Eq. B13) is that a detailed-balance dynamics cannot have a global irreversible property.

Therefore, the total rate function of Eq. (B5) reduces to the sum of the even contributions of the links

$$\begin{aligned}&I_{2.5}[ \rho (.) ; a(.,.) ; j(.,.) ] \nonumber \\&\quad = \sum _{C } \sum _{C' > C} I^{[C',C]\mathrm{{Even}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ]. \nonumber \\ \end{aligned}$$

(B14)

As a consequence, when the empirical density $\rho (.)$ and the empirical activity a(., .) are given, any configuration of the empirical currents $j(C',C)$ that satisfies the stationary constraints of Eq. (B3) has the same rate function as the configuration with the reversed empirical currents $(-j(C',C))$ that also satisfies the stationary constraints

$$\begin{aligned} I_{2.5}[ \rho (.) ; a(.,.) ; j(.,.) ] = I_{2.5}[ \rho (.) ; a(.,.) ; -j(.,.) ]. \nonumber \\ \end{aligned}$$

(B15)

1.3 B.3 Explicit contraction over the currents towards Level 2.25 for the density $\rho (C)$ and the activity $a(C',C)$

The behavior of the even contribution of Eq. (B8) with respect to the current $j(C',C) \in ]- a(C',C),+a(C',C)[$ can be analyzed as follows : the first partial derivative with respect to the current $j(C',C)$

$$\begin{aligned}&\frac{ \partial I^{[C',C]\mathrm{{Even}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ] }{ \partial j (C',C)}\nonumber \\&\quad = \frac{ 1 }{2} \ln \left( \frac{ a(C',C) +j(C',C) }{ a(C',C) -j(C',C) } \right) \end{aligned}$$

(B16)

is of the sign of the current $j(C',C)$, while the second partial derivative remains positive

$$\begin{aligned}&\frac{ \partial ^2 I^{[C',C]\mathrm{{Even}}}_{2.5}[ \rho (C') ; \rho (C) ; a(C',C) ; j(C',C) ] }{ \partial j^2 (C',C)}\nonumber \\&\quad = \frac{ a(C',C) }{ a^2(C',C) -j^2(C',C) } >0. \end{aligned}$$

(B17)

Therefore, the vanishing of the empirical currents on all the links $(C',C)$

$$\begin{aligned} j^\mathrm{{opt}}(C',C)=0 \end{aligned}$$

(B18)

allows to minimize the rate function $I_{2.5}[ \rho (.) ; a(.,.) ; j(.,.) ]$ of Eq. (B14), while the stationary constraints of Eq. (B3) are trivially satisfied. The physical meaning is that for any given empirical density $\rho (.)$ and any given empirical activity a(., .), a detailed-balance dynamics prefers to remain detailed-balance even at the empirical level via the vanishing of all the link empirical currents (Eq. B18).

As a consequence, this optimization over the current of the Level 2.5 of Eq. (B4) yields the large deviations for the joint probability of the empirical density $\rho (.)$ and of the activity a(., .), that we will call the Level 2.25 in the present paper (just to mean that it is between the Level 2.5 described above and the Level 2 that will be described in the next subsection)

$$\begin{aligned}&P^{[2.25]}_{T}[ \rho (.) ; a(.,.) ] \mathop {\propto }_{T \rightarrow +\infty } \delta \left( \sum _C \rho (C) - 1 \right) \nonumber \\&e^{- T I_{2.25}[ \rho (.) ; a(.,.) ], } \end{aligned}$$

(B19)

where the rate function $I_{2.25}[ \rho (.) ; a(.,.) ]$ at Level 2.25 is obtained from the rate function at Level 2.5 of Eqs. (B5) and (B6) when all the empirical currents vanish $j^\mathrm{{opt}}(C',C)=0=0$ (Eq. B18)

$$\begin{aligned}&I_{2.25}[ \rho (.) ; a(.,.) ] = I_{2.5}[ \rho (.) ; a(.,.) ; j^\mathrm{{opt}}(.,.)=0 ] \nonumber \\&= \sum _{C } \sum _{C' > C} \left[ \frac{a(C',C) }{2} \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C) \rho (C) W(C,C') \rho (C')} \right) \right. \nonumber \\&\quad \left. - a(C',C) + W(C',C) \rho (C) + W(C,C') \rho (C') \right] . \ \ \end{aligned}$$

(B20)

1.4 B.4 Explicit contraction of the Level 2.25 over the activity a(., .) towards the Level 2 for the density $\rho (.)$

The optimization of the rate function at Level 2.25 of Eq. (B20) over the empirical activities $a(C',C)$

$$\begin{aligned}&0 = \frac{\partial I_{2.25}[ \rho (.) ; a(.,.) ] }{ \partial a(C',C) }\nonumber \\&\quad = \frac{1 }{2} \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C) \rho (C) W(C,C') \rho (C')} \right) \qquad \end{aligned}$$

(B21)

leads to the optimal values

$$\begin{aligned} a^\mathrm{{opt}}(C',C) = 2 \sqrt{ W(C',C) \rho (C) W(C,C') \rho (C')} \end{aligned}$$

(B22)

that can be plugged into Eq. (B20) to obtain the rate function at Level 2

$$\begin{aligned}&I_{2}[ \rho (.) ] = I_{2.25}[ \rho (.) ; a^\mathrm{{opt}}(.,.) ]\nonumber \\&\quad = \sum _{C } \sum _{C'> C} \left[ W(C',C) \rho (C) + W(C,C') \rho (C') \right. \nonumber \\&\qquad \left. - 2 \sqrt{ W(C',C) \rho (C) W(C,C') \rho (C')} \right] \nonumber \\&= \sum _{C } \sum _{C' > C} \left[ \sqrt{ W(C',C) \rho (C) } - \sqrt{ W(C,C') \rho (C') } \right] ^2 \nonumber \\ \end{aligned}$$

(B23)

that will govern the large deviations properties of the probability of the empirical density $\rho (.)$ alone

$$\begin{aligned} P^{[2]}_{T}[ \rho (.) ] \mathop {\propto }_{T \rightarrow +\infty } \delta \left( \sum _C \rho (C) - 1 \right) e^{- T I_{2}[ \rho (.) ]. } \end{aligned}$$

(B24)

The fact that the Level 2 is closed and explicit for detailed-balance Markov dynamics is well-known since the works of Donsker and Varadhan [99,100,101,102].

1.5 B.5 Contraction of the Level 2.25 over the density $\rho (.)$ to obtain the rate function for the activity a(., .)

A natural question is now whether on can contract the Level 2.25 of Eq. (B19) over the empirical density $\rho (.)$ to obtain the rate function I[a(., .) ] that governs the large deviations properties of the activities a(., .) alone

$$\begin{aligned} P_{T}[ a(.,.) ] \mathop {\propto }_{T \rightarrow +\infty } e^{- T I[ a(.,.) ]. } \end{aligned}$$

(B25)

It is useful to introduce the following notations for the total rate $W^\mathrm{{out}}(C)$ out of the configuration C:

$$\begin{aligned} W^\mathrm{{out}}(C) \equiv \sum _{C' \ne C} W(C',C) \end{aligned}$$

(B26)

and for the total activity of the links connected to the configuration C

$$\begin{aligned} a^\mathrm{{tot}}(C) \equiv \sum _{C' \ne C} a(C',C) \end{aligned}$$

(B27)

Using the symmetry in $(C,C')$ of the activity $a(C',C)=a(C,C')$, the rate function at Level 2.25 of Eq. (B20) can be rewritten using the notations of Eqs. (B26) and (B27) as

$$\begin{aligned}&I_{2.25}[ \rho (.) ; a(.,.) ] = \frac{1}{2} \sum _{C } \sum _{C' \ne C}\nonumber \\&\qquad \frac{a(C',C) }{2} \left[ \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C)W(C,C') } \right) \right. \nonumber \\&\left. - \ln (\rho (C) ) - \ln (\rho (C') ) \right] \nonumber \\&\qquad + \frac{1}{2} \sum _{C } \sum _{C' \ne C} \left[ - a(C',C) \right. \nonumber \\&\qquad \left. + W(C',C) \rho (C) + W(C,C') \rho (C') \right] \nonumber \\&\quad = \sum _{C } \sum _{C' \ne C} \frac{a(C',C) }{4} \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C)W(C,C') } \right) \nonumber \\&\qquad - \sum _{C } \frac{a^\mathrm{{tot}}(C) }{2} \ln (\rho (C) ) - \sum _{C } \frac{a^\mathrm{{tot}}(C) }{2}\nonumber \\&\qquad + \sum _{C } W^\mathrm{{out}}(C) \rho (C). \ \ \ \end{aligned}$$

(B28)

To optimize this rate function at Level 2.25 in the presence of the normalization constraint for the density (Eq. B19), we consider the following Lagrangian involving the Lagrange multiplier $\omega$:

$$\begin{aligned} \Upsilon [ \rho (.) ; a_.(.) ]&\equiv I_{2.25}[ \rho (.) ; a_.(.) ] + \omega \left( 1- \sum _C \rho (C) \right) \nonumber \\&= \sum _{C } \sum _{C' \ne C} \frac{a(C',C) }{4} \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C)W(C,C') } \right) \nonumber \\&\quad + \sum _{C } \left( - \frac{a^\mathrm{{tot}}(C) }{2} - \frac{a^\mathrm{{tot}}(C) }{2} \ln (\rho (C) )\right. \nonumber \\&\left. \quad + \left[ W^\mathrm{{out}}(C) -\omega \right] \rho (C) \right) + \omega . \end{aligned}$$

(B29)

The optimization of this Lagrangian over the empirical densities $\rho (C)$

$$\begin{aligned} 0= \frac{ \partial \Upsilon [ \rho (.) ; a_.(.) ] }{ \partial \rho (C)} = - \frac{a^\mathrm{{tot}}(C) }{2 \rho (C) } + \left[ W^\mathrm{{out}}(C) -\omega \right] \nonumber \\ \end{aligned}$$

(B30)

yields the optimal values

$$\begin{aligned} \rho ^\mathrm{{opt}}(C) = \frac{a^\mathrm{{tot}}(C) }{2 \left[ W^\mathrm{{out}}(C) -\omega \right] , } \end{aligned}$$

(B31)

where the Lagrange multiplier $\omega$ has to be smaller than the total rate $W^\mathrm{{out}}(C)$ out of any configuration C and has to be chosen to satisfy the normalization constraint

$$\begin{aligned} 1= \sum _C \rho ^\mathrm{{opt}}(C) = \sum _C \frac{a^\mathrm{{tot}}(C) }{2 \left[ W^\mathrm{{out}}(C) -\omega \right] . } \end{aligned}$$

(B32)

The rate function I[a(., .) ] that governs the large deviations properties of Eq. (B25) for the activities a(., .) alone corresponds to the value of the Lagrangian of Eq. (B29) when the empirical density $\rho (.)$ takes its optimal value $\rho ^\mathrm{{opt}}(.)$ of Eq (B31) satisfying the normalization constraint of Eq. (B32)

$$\begin{aligned} I[ a(.,.) ]&\equiv \Upsilon [ \rho ^\mathrm{{opt}}(.) ; a(.,.) ] \nonumber \\&= \sum _{C } \sum _{C' \ne C} \frac{a(C',C) }{4} \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C)W(C,C') } \right) \nonumber \\&\quad + \sum _{C } \left( - \frac{a^\mathrm{{tot}}(C) }{2} - \frac{a^{tot}(C) }{2} \ln (\rho ^\mathrm{{opt}}(C) )\right. \nonumber \\&\quad \left. + \left[ W^\mathrm{{out}}(C) -\omega \right] \rho ^\mathrm{{opt}}(C) \right) + \omega \nonumber \\&= \sum _{C } \sum _{C' \ne C} \frac{a(C',C) }{4} \ln \left( \frac{ a^2(C',C) }{ 4 W(C',C)W(C,C') } \right) \nonumber \\&\quad - \sum _{C } \frac{a^\mathrm{{tot}}(C) }{2} \ln \left( \frac{a^\mathrm{{tot}}(C) }{2 \left[ W^\mathrm{{out}}(C) -\omega \right] } \right) + \omega \nonumber \\ \end{aligned}$$

(B33)

with the notations of Eqs. (B26) and (B27). However, this rate function remains somewhat implicit, since the Lagrange multiplier $\omega$ is defined via Eq. (B32). To see more clearly the physical meaning, one can use the equilibrium state $P_\mathrm{{eq}}(C)$ and the equilibrium activities

$$\begin{aligned} A_\mathrm{{eq}}(C',C)= & {} 2 W(C',C) P_\mathrm{{eq}}(C) = 2 W(C,C') P_\mathrm{{eq}}(C') \nonumber \\= & {} A_\mathrm{{eq}}(C,'C) \end{aligned}$$

(B34)

with the corresponding total equilibrium activities of Eq. (B27)

$$\begin{aligned} A^\mathrm{{tot}}_\mathrm{{eq}}(C) = \sum _{C' \ne C} A_\mathrm{{eq}}(C',C) = 2 W^\mathrm{{out}}(C) P_\mathrm{{eq}}(C) \nonumber \\ \end{aligned}$$

(B35)

to rewrite the rates as

$$\begin{aligned} W(C',C)&= \frac{A_\mathrm{{eq}}(C',C) }{ 2 P_\mathrm{{eq}}(C) } \nonumber \\ W^\mathrm{{out}}(C)&= \frac{A^\mathrm{{tot}}_\mathrm{{eq}}(C) }{ 2 P_\mathrm{{eq}}(C) }. \end{aligned}$$

(B36)

Then, Eq. (B32) for the Lagrange multiplier $\omega$ reads

$$\begin{aligned} 1= \sum _C P_\mathrm{{eq}}(C) \ \frac{a^\mathrm{{tot}}(C) }{ A^\mathrm{{tot}}_\mathrm{{eq}}(C) - 2 \omega P_\mathrm{{eq}}(C), } \end{aligned}$$

(B37)

while the rate function of Eq. (B33) becomes

$$\begin{aligned} I[ a(.,.) ]&= \sum _{C } \sum _{C' \ne C} \frac{a(C',C) }{4} \ln \left( \frac{ a^2(C',C) }{ A^2_\mathrm{{eq}}(C',C) } \right) \nonumber \\&\qquad - \sum _{C } \frac{a^\mathrm{{tot}}(C) }{2} \ln \left( \frac{a^\mathrm{{tot}}(C) }{ \left[ A^\mathrm{{tot}}_\mathrm{{eq}}(C) - 2 \omega P_\mathrm{{eq}}(C) \right] } \right) \nonumber \\&\qquad + \omega , \end{aligned}$$

(B38)

where it is now obvious that the value $\omega =0$ is associated with the equilibrium where the rate function vanishes.

1.6 B.6 Comparison with the simplifications for the large deviations of detailed-balance diffusion processes

As a final remark, it is useful to mention the similar simplifications for the large deviation properties of detailed-balance diffusion processes despite some technical differences.

1.6.1 a Reminder on the Level 2.5 for diffusion processes

For diffusion processes described by the Fokker–Planck equation in the force field ${\mathbf {F}}({\mathbf {x}})$ in dimension d, with diffusion coefficient $D({\mathbf {x}})$

$$\begin{aligned} \frac{ \partial P_t({\mathbf {x}}) }{\partial t} = - {\mathbf { \nabla }} . \left[ P_t({\mathbf { x}} ) {\mathbf { F}}({\mathbf { x}} ) -D ({\mathbf { x}}) {\mathbf { \nabla }} P_t({\mathbf { x}}) \right] , \end{aligned}$$

(B39)

the large deviations at Level 2.5 involve the empirical density

$$\begin{aligned} \rho ({\mathbf {x}}) \equiv \frac{1}{T} \int _0^T \mathrm{{d}}t \ \delta ^{(d)} ( {\mathbf {x}}(t)- {\mathbf {x}}) \end{aligned}$$

(B40)

satisfying the normalization

$$\begin{aligned} \int d^d {\mathbf {x}} \ \rho ({\mathbf {x}})&= 1 \end{aligned}$$

(B41)

and the empirical current $\mathbf {j}({\mathbf {x}})$

$$\begin{aligned} \mathbf {j}({\mathbf {x}}) \equiv \frac{1}{T} \int _0^T \mathrm{{d}}t \ \frac{d {\mathbf {x}}(t)}{\mathrm{{d}}t} \delta ^{(d)}( {\mathbf {x}}(t)- {\mathbf {x}}) \end{aligned}$$

(B42)

that should be divergence-free

$$\begin{aligned} {\mathbf {\nabla }} . \mathbf {j}({\mathbf {x}}) =0 \end{aligned}$$

(B43)

to ensure the stationarity.

The joint distribution of the empirical density $\rho (.)$ and the empirical current $\mathbf {j}({\mathbf {x}})$ follows the following large deviation form [24, 25, 29, 30, 33, 43, 45,46,47]:

$$\begin{aligned}&P^{[2.5]}_T[ \rho (.), \mathbf {j}(.)] \mathop {\simeq }_{T \rightarrow +\infty } \delta \left( \int d^d {\mathbf {x}} \rho ({\mathbf {x}}) -1 \right) \nonumber \\&\quad \left[ \prod _{{\mathbf {x}} } \delta \left( {\mathbf {\nabla }} . \mathbf {j}({\mathbf {x}}) \right) \right] e^{- \displaystyle I_{2.5}[ \rho (.), \mathbf {j}(.)] }, \end{aligned}$$

(B44)

where the constitutive constraints have been discussed in Eqs. (B41) and (B43), while the rate function is simply Gaussian with respect to the empirical current $\mathbf {j}({\mathbf {x}})$

$$\begin{aligned}&I_{2.5}[ \rho (.), \mathbf {j}(.)] = \int \frac{d^d {\mathbf { x}}}{ 4 D ({\mathbf { x}}) \rho ({\mathbf { x}}) } \nonumber \\&\quad \times \left[ \mathbf { j}({\mathbf { x}}) - \rho ({\mathbf { x}}) {\mathbf { F}}({\mathbf { x}})+D ({\mathbf { x}}) {\mathbf { \nabla }} \rho ({\mathbf { x}}) \right] ^2. \end{aligned}$$

(B45)

As a consequence, the decomposition into even and odd contributions with respect to the current $\mathbf {j}({\mathbf {x}})$

$$\begin{aligned}&I_{2.5}[ \rho (.), \mathbf {j}(.)]\nonumber \\&\quad = I^\mathrm{{Even}}_{2.5}[ \rho (.), \mathbf {j}(.)] + I^\mathrm{{Odd}}_{2.5}[ \rho (.), \mathbf {j}(.)] \end{aligned}$$

(B46)

involves the even contribution

$$\begin{aligned}&I^\mathrm{{Even}}_{2.5}[ \rho (.), \mathbf { j}(.)] = \int \frac{d^d {\mathbf { x}}}{ 4 D ({\mathbf { x}}) \rho ({\mathbf { x}}) }\nonumber \\&\quad \times \,\left( \mathbf { j}^2({\mathbf { x}}) + \left[ - \rho ({\mathbf { x}}) {\mathbf { F}}({\mathbf { x}})+D ({\mathbf { x}}) {\mathbf { \nabla }} \rho ({\mathbf { x}}) \right] ^2 \right) , \end{aligned}$$

(B47)

while the odd contribution is linear with respect to the current $\mathbf { j}({\mathbf {x}})$

$$\begin{aligned} I^\mathrm{{Odd}}_{2.5}[ \rho (.), \mathbf {j}(.)] = \frac{1}{2} \int d^d {\mathbf {x}} \ \mathbf {j}({\mathbf {x}}) . \left[ - \frac{ {\mathbf {F}}({\mathbf {x}})}{ D ({\mathbf {x}})} + {\mathbf {\nabla }} \ln (\rho ({\mathbf {x}}) ) \right] . \nonumber \\ \end{aligned}$$

(B48)

1.6.2 Simplifications when diffusion processes satisfy Detailed-Balance

For the Fokker–Planck of Eq. (B39), the detailed-balance condition corresponds to the vanishing of the steady current

$$\begin{aligned} 0= P_\mathrm{{eq}}({\mathbf {x}} ) {\mathbf {F}}({\mathbf {x}} ) -D ({\mathbf {x}}) {\mathbf {\nabla }} P_\mathrm{{eq}}({\mathbf {x}}) \end{aligned}$$

(B49)

in the equilibrium state $P_\mathrm{{eq}}({\mathbf {x}} )$ in the potential $U({\mathbf {x}})$ at inverse temperature $\beta$

$$\begin{aligned} P_\mathrm{{eq}}({\mathbf {x}} ) = \frac{e^{- \beta U({\mathbf {x}})}}{Z}, \end{aligned}$$

(B50)

i.e., the force ${\mathbf {F}}({\mathbf {x}} )$ should be of the form

$$\begin{aligned} {\mathbf {F}}({\mathbf {x}} ) = D ({\mathbf {x}}) {\mathbf {\nabla }} \ln ( P_\mathrm{{eq}}({\mathbf {x}}) ) = - \beta D ({\mathbf {x})} {\mathbf {\nabla }} U({\mathbf {x}}). \end{aligned}$$

(B51)

Plugging this detailed-balance force into Eq. (B48), one obtains after an integration by parts and using the stationarity constraint of Eq. (B43) that the odd contribution of Eq. (B48) vanishes

$$\begin{aligned} I^\mathrm{{Odd}}_{2.5}[ \rho (.), \mathbf {j}(.)]&= \frac{1}{2} \int d^d {\mathbf {x}} \ \mathbf {j}({\mathbf {x}}) . {\mathbf {\nabla }} \left[ \beta U({\mathbf {x}}) + \ln (\rho ({\mathbf {x}}) ) \right] \nonumber \\&= - \frac{1}{2} \int d^d {\mathbf {x}} \ \left[ \beta U({\mathbf {x}}) + \ln (\rho ({\mathbf {x}}) ) \right] {\mathbf {\nabla }} . \mathbf {j}({\mathbf {x}}) =0. \nonumber \\ \end{aligned}$$

(B52)

As already mentioned after Eq. (B13) concerning the analog property for Markov jump processes, the physical meaning of this vanishing contribution is that a detailed-balance dynamics cannot have a global irreversible property.

Therefore, the rate function at Level 2.5 of Eq. (B46) reduces to the even contribution of Eq. (B47) with the force of Eq. (B49)

$$\begin{aligned}&I_{2.5}[ \rho (.), \mathbf {j}(.)] = I^\mathrm{{Even}}_{2.5}[ \rho (.), \mathbf {j}(.)]\nonumber \\&\quad = \frac{1}{4} \int d^d {\mathbf {x}} \ D ({\mathbf {x}}) \rho ({\mathbf {x}}) \left( \frac{ \mathbf {j}^2({\mathbf {x}}) }{ D^2 ({\mathbf {x}}) \rho ^2({\mathbf {x}})}\right. \nonumber \\&\qquad \left. + \left[ \beta {\mathbf {\nabla }} U({\mathbf {x}})+ {\mathbf {\nabla }} \ln ( \rho ({\mathbf {x}}) ) \right] ^2 \right) . \end{aligned}$$

(B53)

As a consequence, when the empirical density $\rho (.)$ is given, any configuration of the empirical current $\mathbf {j}(.)$ that satisfies the stationary constraint of Eq. (B43) has the same rate function as the configuration with the reversed empirical current $\mathbf {j}(.)$ that also satisfies the stationary constraint

$$\begin{aligned} I_{2.5}[ \rho (.), \mathbf {j}(.)] = I_{2.5}[ \rho (.), - \mathbf {j}(.)]. \end{aligned}$$

(B54)

When the empirical density $\rho ( {\mathbf {x}})$ is given, the rate function of Eq. (B53) is minimized when the empirical current vanishes everywhere

$$\begin{aligned} \mathbf {j}_{opt} ({\mathbf {x}}) = \mathbf {0}, \end{aligned}$$

(B55)

while the stationarity constraint of Eq. (B43) is trivially satisfied. As already mentioned after Eq. (B18) concerning the analog property for Markov jump processes, the physical meaning is that for any given empirical density $\rho (.)$, a detailed-balance diffusion process prefers to remain detailed-balance even at the empirical level via the vanishing of the empirical current everywhere (Eq. B55).

As a consequence, the contraction of the Level 2.5 of Eq. (B44) over the empirical current $\mathbf {j}({\mathbf {x}})$ is explicit via the optimal solution of Eq. (B55) that leads to the Level 2 for the empirical density $\rho ( {\mathbf {x}})$ alone

$$\begin{aligned} P^{[2]}_T[ \rho (.)] \mathop {\simeq }_{T \rightarrow +\infty } \delta \left( \int d^d {\mathbf {x}} \rho ({\mathbf {x}}) -1 \right) e^{- \displaystyle I_{2}[ \rho (.)] } \end{aligned}$$

(B56)

with the rate function at Level 2 obtained from the rate function at Level 2.5 of Eq. (B53) for vanishing current ${\mathbf {j}}_{opt} ({\mathbf {x}}) = {\mathbf {0}}$

$$\begin{aligned}&I_{2}[ \rho (.)] = I_{2.5}[ \rho (.), \mathbf {j}(.)= \mathbf {0}]\nonumber \\&\quad =\frac{1}{4} \int d^d {\mathbf {x}} \ D ({\mathbf {x}}) \rho ({\mathbf {x}}) \left[ \beta {\mathbf {\nabla }} U({\mathbf {x}})+ {\mathbf {\nabla }} \ln ( \rho ({\mathbf {x}}) ) \right] ^2. \nonumber \\ \end{aligned}$$

(B57)

So here, the contraction of the Level 2.5 over the current gives directly the Level 2 for the empirical density, since there are no ’activity degrees of freedom’ in diffusion processes, in contrast to the Markov jump processes described previously.

Appendix C: Large deviations in the space of the $2^N$ configurations of the random soft East model

In this Appendix, the large deviations at various levels for Detailed-Balance Markov Jump processes summarized in the previous Appendix are applied in the space of the $2^N$ configurations of the random soft East model (Eq. 14).

1.1 C.1 Application of Level 2.5 in the space of the $2^N$ configurations of the random soft East model

For a trajectory $\{ S_1(t),...,S_N(t)\}$ of the N spins over the large time-window $0 \le t \le T$, the empirical time-averaged density of Eq. (A17)

$$\begin{aligned} \rho (S_1,...,S_N)&\equiv \frac{1}{T} \int _0^T \mathrm{{d}}t \ \prod _{n=1}^N \delta _{S_n(t),S_n} \end{aligned}$$

(C1)

satisfies the normalization of Eq. (A18)

$$\begin{aligned} \sum _{S_1=\pm } ... \sum _{S_N=\pm } \rho (S_1,...,S_N) = 1. \end{aligned}$$

(C2)

The empirical flows of Eq. (A19) associated with the flip rates $w_{i}^{S_i} (S_{i-1})$ of the model read

$$\begin{aligned}&q_{i}^{S_i} (S_1,..,S_{i-1} ; S_{i+1} ,..,S_N) \equiv \frac{1}{T}\nonumber \\&\sum _{t \in [0,T] : \begin{array}{c} S_i(t)=S_i \\ S_i(t^+)= - S_i \end{array}}\nonumber \\&\left[ \prod _{n=1}^{i-1} \delta _{S_n(t),S_n} \right] \left[ \prod _{p=i+1}^{N} \delta _{S_p(t),S_p} \right] . \end{aligned}$$

(C3)

The stationarity constraint of Eq. (A22) for the configuration $(S_1,...,S_N)$ reads

$$\begin{aligned} 0 = \sum _{i=1 }^{N} \left[ q_{i}^{-S_i} (S_1,..,S_{i-1} ; S_{i+1} ,..,S_N)\right. \nonumber \\\left. - q_{i}^{S_i} (S_1,..,S_{i-1} ; S_{i+1} ,..,S_N) \right] . \end{aligned}$$

(C4)

The rate function of Eq. (A26) that governs the large deviations at Level 2.5 of Eq. (A27) reads

$$\begin{aligned}&I_{2.5}[ \rho (.) ; q_.^{\pm }(.) ] = \sum _{S_1 = \pm } \sum _{S_2 = \pm } ... \sum _{S_N = \pm }\nonumber \\&\quad \sum _{i=1}^{N} \bigg [ q_{i}^{S_i} (..,S_{i-1} ; S_{i+1} ,..)\nonumber \\&\ln \left( \frac{q_{i}^{S_i} (..,S_{i-1} ; S_{i+1} ,..) }{ w_{i}^{S_i} (S_{i-1}) \rho (..S_{i-1},S_i,S_{i+1}..) } \right) \nonumber \\&- q_{i}^{S_i} (..,S_{i-1} ; S_{i+1} ,..) + w_{i}^{S_i} (S_{i-1}) \rho (..S_{i-1},S_i,S_{i+1}..) \bigg ]. \nonumber \\ \end{aligned}$$

(C5)

The parametrizations of Eq. (B1) for the empirical flows

$$\begin{aligned}&q_{i}^{+} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&\quad \equiv \frac{a_{i} (..,S_{i-1} ; S_{i+1} ,..) +j_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2} \nonumber \\&q_{i}^{-} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&\quad \equiv \frac{a_{i} (..,S_{i-1} ; S_{i+1} ,..) - j_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2} \nonumber \\ \end{aligned}$$

(C6)

in terms of the activities and the currents of Eq. (B2)

$$\begin{aligned}&a_{i} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&\quad \equiv q_{i}^{+} (..,S_{i-1} ; S_{i+1} ,..) + q_{i}^{-} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&j_{i} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&\quad \equiv q_{i}^{+} (..,S_{i-1} ; S_{i+1} ,..) - q_{i}^{-} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\ \end{aligned}$$

(C7)

allows to rewrite the stationarity constraints of Eq. (C4) in terms of the currents only (Eq. B3)

$$\begin{aligned} 0 = \sum _{i=1 }^{N} j_{i} (S_1,..,S_{i-1} ; S_{i+1} ,..,S_N). \end{aligned}$$

(C8)

1.2 C.2 Application of Level 2.25 in the space of the $2^N$ configurations of the random soft East model

As explained in detail in Appendix 1, the detailed-balance property satisfied by the rates allows to make the explicit contraction of the Level 2.5 over the empirical currents via the simple optimal solution where the empirical currents on all the links (Eq. B18)

$$\begin{aligned} j^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..)=0 \end{aligned}$$

(C9)

and lead to the Level 2.25 of Eq. (B19) for the empirical density and the empirical activities

$$\begin{aligned}&P^{[2.25]}_{T}[ \rho (.) ; a_.(.) ] \mathop {\propto }_{T \rightarrow +\infty } \delta \left( \sum _{S_1=\pm } ... \sum _{S_N=\pm } \rho (S_1,...,S_N) - 1 \right) \nonumber \\&\quad e^{- T I_{2.25}[ \rho (.) ; a(.) ] } \end{aligned}$$

(C10)

with the rate function of Eq. (B20)

$$\begin{aligned}&I_{2.25}[ \rho (.) ; a_.(.) ] = \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm } \right] \bigg [ \frac{a_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2} \nonumber \\&\quad \ln \left( \frac{ a^2_{i} (..,S_{i-1} ; S_{i+1} ,..) }{ 4 w_{i}^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w_{i}^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..)} \right) \nonumber \\&\qquad - a_{i} (..,S_{i-1} ; S_{i+1} ,..) + w_{i}^{+} (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..,)\nonumber \\&\qquad + w_{i}^{-}(S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..) \bigg ]. \end{aligned}$$

(C11)

1.3 C.3 Application of Level 2 in the space of the $2^N$ configurations of the random soft East model

Finally, the Level 2 for the empirical density alone of Eq. (B24)

$$\begin{aligned} P^{[2]}_{T}[ \rho (.) ] \mathop {\propto }_{T \rightarrow +\infty } \delta \left( \sum _{S_1=\pm } ... \sum _{S_N=\pm } \rho (S_1,...,S_N) - 1 \right) \nonumber \\ e^{- T I_{2}[ \rho (.) ] }\nonumber \\ \end{aligned}$$

(C12)

involves the rate function at Level 2 of Eq. (B23)

$$\begin{aligned}&I_{2}[ \rho (.) ] = \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm } \right] \nonumber \\&\left[ \sqrt{ w_{i}^{+} (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..,) } \right. \nonumber \\&\left. - \sqrt{ w_{i}^{-}(S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..) } \right] ^2. \end{aligned}$$

(C13)

Appendix D: Random soft East model: contraction of the global Level 2.25 towards the local Level 2.25

In this Appendix, we describe the explicit contraction from the Level 2.25 of Eq. (C10) in the space of the $2^N$ configurations of the random soft East model towards the Level 2.25 for the local densities and the local activities, as given by Eq. (31) of the main text.

1.1 D.1 Local empirical observables from global empirical observables in the space of the $2^N$ configurations

The empirical 2-spin density $\rho _{i-1,i}^{S_{i-1},S_{i}}$ of Eq. (17) can be obtained from the configuration empirical density $\rho (S_1,...,S_N)$ of Eq. (C1) by summing over the $(N-2)$ other spins $n \ne (i-1,i)$

$$\begin{aligned} \rho _{i-1,i}^{S_{i-1},S_{i}} = \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \rho (S_1,...,S_N). \nonumber \\ \end{aligned}$$

(D1)

Similarly, the local empirical activities $a_i(S_{i-1})$ of Eq. (23) can be obtained from the configuration activities $a_{i} (S_1,..,S_{i-1} ; S_{i+1} ,..,S_N)$ of Eq. (C7) by summing over the $(N-2)$ other spins $n \ne (i-1,i)$

$$\begin{aligned} a_i(S_{i-1}) = \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] a_{i} (..,S_{i-1} ; S_{i+1} ,..). \nonumber \\ \end{aligned}$$

(D2)

1.2 D.2 Contraction of the global Level 2.25 with constraints fixing the local empirical observables

To optimize the rate function at Level 2.25 of Eq. (C11) over the configuration empirical density $\rho (S_1,...,S_N)$ and over the configuration empirical activities $a_{i} (..,S_{i-1} ; S_{i+1} ,..)$ in the presence of the constraints of Eqs. (D1) and (D2), we consider the following Lagrangian involving the Lagrange multipliers $[\omega _{i-1,i}^{\pm ,\pm } ; \lambda _i(\pm )]$ for $i=1,..,N$:

$$\begin{aligned}&\Upsilon [ \rho (.) ; a_.(.) ] \equiv I_{2.25}[ \rho (.) ; a_.(.) ]\nonumber \\&\quad + \sum _{i=1}^N \sum _{S_{i-1}=\pm }\sum _{S_{i}=\pm } \omega _{i-1,i}^{S_{i-1},S_{i}} \left( \rho _{i-1,i}^{S_{i-1},S_{i}} - \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \right. \nonumber \\&\quad \left. \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \rho (S_1,...,S_N) \right) \nonumber \\&\quad + \sum _{i=1}^N \sum _{S_{i-1}=\pm }\lambda _i(S_{i-1})\nonumber \\&\quad \left( a_i(S_{i-1}) - \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \right. \nonumber \\&\quad \left. \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] a_{i} (..,S_{i-1} ; S_{i+1} ,..) \right) . \end{aligned}$$

(D3)

The explicit rate function at Level 2.25 of Eq. (C11) yields

$$\begin{aligned}&\Upsilon [ \rho (.) ; a_.(.) ] = \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm } \right] \bigg [ \frac{a_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2} \ln \left( \frac{ a^2_{i} (..,S_{i-1} ; S_{i+1} ,..) }{ 4 w_{i}^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w_{i}^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..)} \right) \nonumber \\&\quad - a_{i} (..,S_{i-1} ; S_{i+1} ,..) + w_{i}^{+} (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..,)\nonumber \\&\quad + w_{i}^{-}(S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..) \bigg ] + \sum _{i=1}^N \sum _{S_{i-1}=\pm }\sum _{S_{i}=\pm } \omega _{i-1,i}^{S_{i-1},S_{i}} \left( \rho _{i-1,i}^{S_{i-1},S_{i}} - \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \rho (S_1,...,S_N) \right) \nonumber \\&\quad + \sum _{i=1}^N \sum _{S_{i-1}=\pm }\lambda _i(S_{i-1}) \left( a_i(S_{i-1}) - \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] a_{i} (..,S_{i-1} ; S_{i+1} ,..) \right) . \end{aligned}$$

(D4)

The optimization of this Lagrangian over the configuration activities $a_{i} (..,S_{i-1} ; S_{i+1} ,..)$

$$\begin{aligned} 0= \frac{ \partial \Upsilon [ \rho (.) ; a_.(.) ] }{ \partial a_{i} (..,S_{i-1} ; S_{i+1} ,..)} = \frac{1 }{2} \ln \left( \frac{ a^2_{i} (..,S_{i-1} ; S_{i+1} ,..) }{ 4 w_{i}^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w_{i}^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..)} \right) - \lambda _i(S_{i-1}) \end{aligned}$$

(D5)

leads to the optimal values

$$\begin{aligned}&a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) = 2 e^{ \lambda _i(S_{i-1})} \sqrt{ w_{i}^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w_{i}^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..) }, \nonumber \\ \end{aligned}$$

(D6)

where the Lagrange multipliers $\lambda _i(S_{i-1})$ have to be chosen to satisfy the corresponding constraints of Eq. (D2)

$$\begin{aligned}&a_i(S_{i-1}) = \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&\quad = 2 e^{ \lambda _i(S_{i-1})} \sqrt{ w_{i}^+ (S_{i-1}) w_{i}^{-} (S_{i-1}) } D_i(S_{i-1}), \end{aligned}$$

(D7)

where we have introduced the notation

$$\begin{aligned}&D_i(S_{i-1}) \equiv \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad \sqrt{ \rho (..,S_{i-1},+, S_{i+1},..) \rho (..,S_{i-1},-, S_{i+1},..). }\nonumber \\ \end{aligned}$$

(D8)

The optimal values of Eq. (D6) can be thus rewritten as

$$\begin{aligned}&a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..)\nonumber \\&\quad = a_i(S_{i-1}) \frac{ \sqrt{ \rho (..,S_{i-1},+, S_{i+1},..) \rho (..,S_{i-1},-, S_{i+1},..) } }{D_i(S_{i-1}). } \nonumber \\ \end{aligned}$$

(D9)

The optimization of the Lagrangian of Eq. (D4) over the configuration density $\rho (S_1,...,S_N)$

(D10)

yields that the optimal values $\rho ^\mathrm{{opt}}(S_1,...,S_N)$ have to satisfy together with the optimal values $a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..)$ of the activities of Eq. (D9)

$$\begin{aligned}&0= \sum _{i=1}^{N} \left[ - \frac{ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) }{2} \right. \nonumber \\&\qquad \left. + \left( w_{i}^{S_i} (S_{i-1}) - \omega _{i-1,i}^{S_{i-1},S_{i}} \right) \rho ^\mathrm{{opt}}(S_1,...,S_N) \right] . \nonumber \\ \end{aligned}$$

(D11)

The simplest way to satisfy this equation for the sum of N terms is to impose the vanishing of each term for $i=1,..,N$

$$\begin{aligned}&0= - \frac{ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) }{2}\nonumber \\&+ \left( w_{i}^{S_i} (S_{i-1}) - \omega _{i-1,i}^{S_{i-1},S_{i}} \right) \rho ^\mathrm{{opt}}(S_1,...,S_N). \nonumber \\ \end{aligned}$$

(D12)

The summation over the spins $S_k=\pm$ for $k =1,..,i-2$ and $k=i+1,..,N$ yields using the constraints of Eqs. (D1) and (D2)

$$\begin{aligned}&0=- \frac{ a_{i} (S_{i-1}) }{2} + \left( w_{i}^{S_i} (S_{i-1}) - \omega _{i-1,i}^{S_{i-1},S_{i}}\right) \rho _{i-1,i}^{S_{i-1},S_{i}},\nonumber \\ \end{aligned}$$

(D13)

so the Lagrange multipliers $\omega _{i-1,i}^{S_{i-1},S_{i}}$ that modify the true rates $w_{i}^{S_i} (S_{i-1})$ to produce the effective rates $\Big ( w_{i}^{S_i} (S_{i-1}) -\omega _{i-1,i}^{S_{i-1},S_{i}} \Big )$ can be computed from the ratios

$$\begin{aligned} \left( w_{i}^{S_i} (S_{i-1}) - \omega _{i-1,i}^{S_{i-1},S_{i}} \right) =\frac{a_{i} (S_{i-1}) }{2 \rho _{i-1,i}^{S_{i-1},S_{i}}}; \end{aligned}$$

(D14)

Equation (D12) then yields the optimal density

$$\begin{aligned} \rho ^\mathrm{{opt}}(S_1,...,S_N) = \rho _{i-1,i}^{S_{i-1},S_{i}} \ \frac{ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) }{ a_{i} (S_{i-1})}\nonumber \\ \end{aligned}$$

(D15)

that can be plugged into Eq. (D8) to obtain with the use of the constraint of Eq. (D2)

$$\begin{aligned}&D_i(S_{i-1}) = \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\sqrt{ \rho ^\mathrm{{opt}} (..,S_{i-1},+, S_{i+1},..) \rho ^\mathrm{{opt}} (..,S_{i-1},-, S_{i+1},..) } \nonumber \\&= \frac{ \sqrt{ \rho _{i-1,i}^{S_{i-1},+} \rho _{i-1,i}^{S_{i-1},-} } }{ a_{i} (S_{i-1}) }\nonumber \\&\quad \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) = \sqrt{ \rho _{i-1,i}^{S_{i-1},+} \rho _{i-1,i}^{S_{i-1},-} }, \end{aligned}$$

(D16)

so that the Lagrange multipliers $\lambda _i(S_{i-1})$ of Eq. (D7) reduce to

$$\begin{aligned} \lambda _i(S_{i-1})= & {} \ln \left( \frac{ a_i(S_{i-1}) }{ 2 \sqrt{ w_{i}^+ (S_{i-1}) w_{i}^{-} (S_{i-1}) \rho _{i-1,i}^{S_{i-1},+} \rho _{i-1,i}^{S_{i-1},-} } } \right) \nonumber \\= & {} \frac{1}{2} \ln \left( \frac{ a^2_i(S_{i-1}) }{ 4 w_{i}^+ (S_{i-1}) w_{i}^{-} (S_{i-1}) \rho _{i-1,i}^{S_{i-1},+} \rho _{i-1,i}^{S_{i-1},-} } \right) . \nonumber \\ \end{aligned}$$

(D17)

The optimal value of the Lagrangian of Eq. (D4) corresponding to the optimal solution $[ \rho ^\mathrm{{opt}}(.) ; a^\mathrm{{opt}}_.(.) ]$ satisfying the constraints reads using Eq. (D17)

$$\begin{aligned}&\Upsilon ^\mathrm{{opt}} \equiv \Upsilon [ \rho ^\mathrm{{opt}}(.) ; a^\mathrm{{opt}}_.(.) ] = \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm }\right] \nonumber \\&\quad \frac{a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2} \ln \left( \frac{ [ a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..)]^2 }{ 4 w_{i}^+ (S_{i-1}) \rho ^\mathrm{{opt}} (..,S_{i-1},+, S_{i+1},..) w_{i}^{-} (S_{i-1}) \rho ^\mathrm{{opt}} (..,S_{i-1},-, S_{i+1},..)} \right) \nonumber \\&\quad + \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm } \right] \bigg [ - a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) + w_{i}^{+} (S_{i-1}) \rho ^\mathrm{{opt}} (..,S_{i-1},+, S_{i+1},..,)\nonumber \\&\quad + w_{i}^{-}(S_{i-1}) \rho ^\mathrm{{opt}}(..,S_{i-1},-, S_{i+1},..) \bigg ] \nonumber \\&\quad = \sum _{i=1}^{N} \sum _{S_{i-1} = \pm } \left[ a_{i} (S_{i-1}) \lambda _i(S_{i-1}) - a_{i} (S_{i-1}) + w_{i}^{+} (S_{i-1}) \rho _{i-1,i}^{S_{i-1},+} + w_{i}^{-}(S_{i-1}) \rho _{i-1,i}^{S_{i-1},-} \right] \nonumber \\&\quad = \sum _{i=1}^{N} \sum _{S_{i-1} = \pm } \left[ \frac{a_{i} (S_{i-1})}{2} \ln \left( \frac{ a^2_i(S_{i-1}) }{ 4 w_{i}^+ (S_{i-1}) w_{i}^{-} (S_{i-1}) \rho _{i-1,i}^{S_{i-1},+} \rho _{i-1,i}^{S_{i-1},-} } \right) \right. \nonumber \\&\quad \left. - a_{i} (S_{i-1}) + w_{i}^{+} (S_{i-1}) \rho _{i-1,i}^{S_{i-1},+} + w_{i}^{-}(S_{i-1}) \rho _{i-1,i}^{S_{i-1},-} \right] . \end{aligned}$$

(D18)

This optimal value $\Upsilon ^\mathrm{{opt}}$ corresponds to the rate function $I_{2.25} [ a_.(.) ; \rho _{.,.}^{.,.}]$ at Level 2.25 for the local activities $a_{i} (S_{i-1})$ and the local densities $\rho _{i-1,i}^{S_{i-1},S_i}$ as given in Eq. (32) of the main text

$$\begin{aligned} \Upsilon ^\mathrm{{opt}} = I_{2.25} [ a_.(.) ; \rho _{.,.}^{.,.}] \end{aligned}$$

(D19)

Appendix E: Pure soft East model: contraction of the global Level 2.25 towards the local Level 2.25

In this Appendix, we describe the explicit contraction from the Level 2.25 of Eq. (C10) in the space of the $2^N$ configurations of the pure soft East model towards the Level 2.25, for the empirical time–space-averaged densities and activities, as given by Eq. (81) of the main text. This contraction is very similar to the contraction described in the previous Appendix.

1.1 E.1 Empirical time–space-averaged observables from observables in the space of the $2^N$ configurations

The empirical time–space-averaged density $\rho ^{S_L,S}$ of two consecutive spins $(S_L,S)$ of Eq. (71) can be obtained from the configuration empirical time-averaged density $\rho (S_1,...,S_N)$ of Eq. (C1) via

$$\begin{aligned}&\rho ^{S_L,S} = \frac{1}{N} \sum _{i=1}^N \left[ \prod _{n=1}^{N} \sum _{S_n=\pm } \right] \nonumber \\&\quad \delta _{S_{i-1},S_L} \delta _{S_{i},S} \ \rho (S_1,...,S_N). \end{aligned}$$

(E1)

Similarly, the empirical time-space-averaged activity $a(S_L)$ of Eq. (77) can be obtained from the configuration empirical activities $a_{i} (S_1,..,S_{i-1} ; S_{i+1} ,..,S_N)$ of Eq. (C7) via

$$\begin{aligned}&a(S_L) =\frac{1}{N} \sum _{i=1}^N \left[ \prod _{n=1}^{i-1} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad \delta _{S_{i-1},S_L} \ a_{i} (..,S_{i-1} ; S_{i+1} ,..). \end{aligned}$$

(E2)

1.2 E.2 Contraction of the global Level 2.25 with constraints fixing the time–space-averaged observables

To optimize the rate function at Level 2.25 of Eq. (C11) over the configuration empirical density $\rho (S_1,...,S_N)$ and over the configuration empirical activities $a_{i} (..,S_{i-1} ; S_{i+1} ,..)$ in the presence of the constraints of Eqs. (E1) and Eq. (E2), we consider the following Lagrangian involving the Lagrange multipliers $[\omega ^{\pm ,\pm } ; \lambda (\pm )]$:

$$\begin{aligned}&\Upsilon [ \rho (.) ; a_.(.) ]\nonumber \\&\quad \equiv I_{2.25}[ \rho (.) ; a_.(.) ] + \sum _{S_L=\pm }\sum _{S=\pm } \omega ^{S_L,S}\nonumber \\&\quad \left( N \rho ^{S_L,S} - \sum _{i=1}^N \left[ \prod _{n=1}^{N} \sum _{S_n=\pm } \right] \right. \nonumber \\&\quad \left. \delta _{S_{i-1},S_L} \delta _{S_{i},S} \rho (S_1,...,S_N) \right) \nonumber \\&\quad + \sum _{S_L=\pm }\lambda (S_L) \left( N a(S_L) - \sum _{i=1}^N \left[ \prod _{n=1}^{i-1} \sum _{S_n=\pm } \right] \right. \nonumber \\&\quad \left. \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \delta _{S_{i-1},S_L} \ a_{i} (..,S_{i-1} ; S_{i+1} ,..) \right) .\nonumber \\ \end{aligned}$$

(E3)

The explicit rate function at Level 2.25 of Eq. (C11) yields

$$\begin{aligned}&\Upsilon [ \rho (.) ; a_.(.) ] = \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm } \right] \bigg [ \frac{a_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2} \nonumber \\&\quad \ln \left( \frac{ a^2_{i} (..,S_{i-1} ; S_{i+1} ,..) }{ 4 w^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..)} \right) \nonumber \\&\quad - a_{i} (..,S_{i-1} ; S_{i+1} ,..) + w^{+} (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..,)\nonumber \\&\quad + w^{-}(S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..) \bigg ] \nonumber \\&\quad + \sum _{S_L=\pm }\sum _{S=\pm } \omega ^{S_L,S} \left( N \rho ^{S_L,S} - \sum _{i=1}^N \left[ \prod _{n=1}^{N} \sum _{S_n=\pm } \right] \right. \nonumber \\&\left. \quad \delta _{S_{i-1},S_L} \delta _{S_{i},S} \ \rho (S_1,...,S_N) \right) \nonumber \\&\quad + \sum _{S_L=\pm }\lambda (S_L) \left( N a(S_L) - \sum _{i=1}^N \left[ \prod _{n=1}^{i-1} \sum _{S_n=\pm } \right] \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \delta _{S_{i-1},S_L} \ a_{i} (..,S_{i-1} ; S_{i+1} ,..) \right) . \end{aligned}$$

(E4)

The optimization of this Lagrangian over the configuration activities $a_{i} (..,S_{i-1} ; S_{i+1} ,..)$

$$\begin{aligned} 0= \frac{ \partial \Upsilon [ \rho (.) ; a_.(.) ] }{ \partial a_{i} (..,S_{i-1} ; S_{i+1} ,..)} = \frac{1 }{2} \ln \left( \frac{ a^2_{i} (..,S_{i-1} ; S_{i+1} ,..) }{ 4 w^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..)} \right) - \lambda (S_{i-1}) \end{aligned}$$

(E5)

leads to the optimal values

$$\begin{aligned}&a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) = 2 e^{ \lambda (S_{i-1})}\nonumber \\&\quad \sqrt{ w^+ (S_{i-1}) \rho (..,S_{i-1},+, S_{i+1},..) w^{-} (S_{i-1}) \rho (..,S_{i-1},-, S_{i+1},..) } \nonumber \\ \end{aligned}$$

(E6)

where the Lagrange multipliers $\lambda (S_L)$ have to be chosen to satisfy the corresponding constraints of Eq. (E2)

$$\begin{aligned}&a(S_L) =\frac{1}{N} \sum _{i=1}^N \left[ \prod _{n=1}^{i-1} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad \delta _{S_{i-1},S_L} \ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&= 2 e^{ \lambda (S_L) } \sqrt{ w^+ (S_L) w^{-} (S_L) } D(S_L), \end{aligned}$$

(E7)

where we have introduced the notation

$$\begin{aligned}&D(S_L) \equiv \frac{1}{N} \sum _{i=1}^N \nonumber \\&\quad \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad \sqrt{ \rho (..S_{i-2},S_L,+, S_{i+1},..) \rho (..,S_{i-2},S_L,-, S_{i+1},..) }. \nonumber \\ \end{aligned}$$

(E8)

The optimal values of Eq. (E6) can be thus rewritten as

$$\begin{aligned}&a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) = a(S_{i-1})\nonumber \\&\quad \frac{ \sqrt{ \rho (..,S_{i-1},+, S_{i+1},..) \rho (..,S_{i-1},-, S_{i+1},..) } }{D(S_{i-1}) }. \nonumber \\ \end{aligned}$$

(E9)

The optimization of the Lagrangian of Eq. (D4) over the configuration density $\rho (S_1,...,S_N)$

$$\begin{aligned} 0= & {} \frac{ \partial \Upsilon [ \rho (.) ; a_.(.) ] }{ \partial \rho (S_1,...,S_N)} \nonumber \\= & {} - \frac{1}{2 \rho (S_1,...,S_N)} \sum _{i=1}^{N} a_{i} (..,S_{i-1} ; S_{i+1} ,..)\nonumber \\&+ \sum _{i=1}^N \left( w^{S_i} (S_{i-1}) - \omega ^{S_{i-1},S_{i}} \right) \end{aligned}$$

(E10)

yields that the optimal values $\rho ^\mathrm{{opt}}(S_1,...,S_N)$ have to satisfy together with the optimal values $a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..)$ of the activities of Eq. (E9)

$$\begin{aligned} 0= & {} \sum _{i=1}^{N} \left[ - \frac{ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) }{2}\right. \nonumber \\&\left. + \left( w^{S_i} (S_{i-1}) - \omega ^{S_{i-1},S_{i}} \right) \rho ^\mathrm{{opt}}(S_1,...,S_N) \right] . \nonumber \\ \end{aligned}$$

(E11)

The simplest way to satisfy this equation for the sum of N terms is to impose the vanishing of each term for $i=1,..,N$

$$\begin{aligned} 0= & {} - \frac{ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) }{2}\nonumber \\&+ \left( w^{S_i} (S_{i-1}) - \omega ^{S_{i-1},S_{i}} \right) \rho ^\mathrm{{opt}}(S_1,...,S_N). \nonumber \\ \end{aligned}$$

(E12)

Let us apply the sum $\left( \frac{1}{N} \sum _{i=1}^N \left[ \prod _{n=1}^{N} \sum _{S_n=\pm } \right] \delta _{S_{i-1},S_L} \delta _{S_{i},S} \right)$ to Eq. (E12) in order to be able to use the constraints of Eqs. (E1) and (E2)

$$\begin{aligned} 0= & {} \frac{1}{N} \sum _{i=1}^N \left[ \prod _{n=1}^{N} \sum _{S_n=\pm } \right] \nonumber \\&\delta _{S_{i-1},S_L} \delta _{S_{i},S} \left[ - \frac{ a_{i} (S_{i-1}) }{2}\right. \nonumber \\&\left. + \left( w^{S_i} (S_{i-1}) - \omega ^{S_{i-1},S_{i}} \right) \rho _{i-1,i}^{S_{i-1},S_{i}} \right] \nonumber \\= & {} - \frac{ a (S_L) }{2} + \left( w^{S} (S_L) - \omega ^{S_L,S} \right) \rho ^{S_L,S}. \end{aligned}$$

(E13)

Therefore, the Lagrange multipliers $\omega ^{S_L,S}$ that modify the true rates $w^{S} (S_L)$ to produce the effective rates $\left( w^{S} (S_L) - \omega ^{S_L,S} \right)$ can be computed from the ratios

$$\begin{aligned} \left( w^{S} (S_L) - \omega ^{S_L,S} \right) = \frac{ a (S_L) }{2 \rho ^{S_L,S}}; \end{aligned}$$

(E14)

Eq. (E12) then yields the optimal density

$$\begin{aligned} \rho ^\mathrm{{opt}}(S_1,...,S_N) = \rho ^{S_{i-1},S_{i}} \ \frac{ a_{i}^\mathrm{{opt}} (..,S_{i-1} ; S_{i+1} ,..) }{ a (S_{i-1})} \nonumber \\ \end{aligned}$$

(E15)

that can be plugged into Eq. (E8) to obtain with the use of the constraint of Eq. (E2)

$$\begin{aligned}&D(S_L) = \frac{1}{N} \sum _{i=1}^N \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] \nonumber \\&\quad \sqrt{ \rho ^\mathrm{{opt}} (..S_{i-2},S_L,+, S_{i+1},..) \rho ^\mathrm{{opt}} (..,S_{i-2},S_L,-, S_{i+1},..) } \nonumber \\&\quad =\frac{ \sqrt{ \rho ^{S_L,+} \rho ^{S_L,-} } }{ N a (S_L)}\nonumber \\&\quad \sum _{i=1}^N \left[ \prod _{n=1}^{i-2} \sum _{S_n=\pm } \right] \nonumber \\&\quad \left[ \prod _{p=i+1}^{N}\sum _{S_p=\pm } \right] a_{i}^\mathrm{{opt}} (..,S_L ; S_{i+1} ,..)\nonumber \\&\quad = \sqrt{ \rho ^{S_L,+} \rho ^{S_L,-} }. \end{aligned}$$

(E16)

Therefore, the Lagrange multipliers $\lambda (S_L)$ of Eq. (E7) reduce to

$$\begin{aligned}&\lambda (S_L) = \ln \left( \frac{ a(S_L) }{ 2 \sqrt{ w^+ (S_L) w^{-} (S_L) \rho ^{S_L,+} \rho ^{S_L,-} } } \right) \nonumber \\&\quad = \frac{1}{2} \ln \left( \frac{ a^2(S_L) }{ 4 w^+ (S_L) w^{-} (S_L) \rho ^{S_L,+} \rho ^{S_L,-} } \right) . \end{aligned}$$

(E17)

The optimal value of the Lagrangian of Eq. (E4) corresponding to the optimal solution $[ \rho ^\mathrm{{opt}}(.) ; a^\mathrm{{opt}}_.(.) ]$ satisfying the constraints reads using Eq. (E17)

$$\begin{aligned}&\Upsilon ^\mathrm{{opt}} \equiv \Upsilon [ \rho ^\mathrm{{opt}}(.) ; a^\mathrm{{opt}}_.(.) ] = \sum _{i=1}^{N} \left[ \prod _{k \ne i} \sum _{S_k = \pm }\right] \nonumber \\&\quad \bigg [ \frac{a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) }{2}\nonumber \\&\quad \ln \left( \frac{ (a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) )^2 }{ 4 w^+ (S_{i-1}) \rho ^\mathrm{{opt}} (..,S_{i-1},+, S_{i+1},..) w^{-} (S_{i-1}) \rho ^\mathrm{{opt}} (..,S_{i-1},-, S_{i+1},..)} \right) - a^\mathrm{{opt}}_{i} (..,S_{i-1} ; S_{i+1} ,..) \nonumber \\&\quad + w^{+} (S_{i-1})\rho ^\mathrm{{opt}} (..,S_{i-1},+, S_{i+1},..,) + w^{-}(S_{i-1}) \rho ^\mathrm{{opt}}(..,S_{i-1},-, S_{i+1},..) \bigg ]\nonumber \\&= \sum _{S_L=\pm } \lambda (S_L) \sum _{i=1}^{N} \left[ \prod _{k\ne (i-1,i)} \sum _{S_k = \pm } \right] a^\mathrm{{opt}}_{i} (..S_{i-2},S_L ;S_{i+1} ,..) - \sum _{S_L=\pm } \sum _{i=1}^{N} \left[ \prod _{k \ne (i-1,i)} \sum _{S_k = \pm } \right] a^\mathrm{{opt}}_{i} (..,S_{i-2},S_L ; S_{i+1} ,..) \nonumber \\&\quad + \sum _{S_L=\pm } \sum _{i=1}^{N} \left[ \prod _{k \ne (i-1,i)} \sum _{S_k = \pm } \right] \bigg [ w^{+} (S_L) \rho ^\mathrm{{opt}} (..,S_{i-2},S_L,+, S_{i+1},..,) + w^{-}(S_L) \rho ^\mathrm{{opt}}(..,S_{i-2},S_L,-, S_{i+1},..) \bigg ] \nonumber \\&\quad = N \sum _{S_L=\pm } \left[ \frac{a(S_L)}{2} \ln \left( \frac{ a^2(S_L) }{ 4 w^+ (S_L) w^{-} (S_L) \rho ^{S_L,+} \rho ^{S_L,-} } \right) - a(S_L) + w^{+} (S_L) \rho ^{S_L,+} + w^{-}(S_L) \rho ^\mathrm{{opt}}(S_L,-) \right] . \end{aligned}$$

(E18)

This optimal value $\Upsilon ^\mathrm{{opt}}$ allows to recover the rate function $\mathcal{I}_{2.25} [ a(.) ; \rho ^{.,.}]$ with respect to the space–time volume (NT) at Level 2.25 for the empirical time–space-averaged activities $a (\pm )$ and the local time–space-averaged densities $\rho ^{S_L,S}$ as given in Eq. (82) of the main text

$$\begin{aligned} \Upsilon ^\mathrm{{opt}} = N \mathcal{I}_{2.25} [ a(.) ; \rho ^{.,.}]. \end{aligned}$$

(E19)