Spin glass polynomial identities from entropic constraints

To introduce the model (originally studied by Viana and Bray in [31]), let us consider N Ising spins σ_i ∈ { + 1, −1}, with i running from 1 to N; σ = (σ₁, ..., σ_N) will denote the complete spin configuration. Let P_ζ be a Poisson random variable of mean ζ, and let {J_ν} be independent and identically distributed (i.i.d.) copies of a random coupling strength variable J with a symmetric distribution. For the sake of simplicity, and without undue loss of generality [21], we will assume J = ±1. The coupling strengths J_ν will determine binary interactions between spins at sites {i_ν}, {j_ν}; the latter are independent identically distributed random variables, with the uniform distribution over 1, ..., N. If there is no external field, the Hamiltonian of the VB model for the dilute mean field spin glass is then

$$\begin{equation} H_N(\sigma , \alpha ; \mathcal {J})= -\sum _{\nu =1}^{P_{\alpha N}} J_\nu \sigma _{i_\nu }\sigma _{j_\nu } ,\quad \alpha \in \mathbb {R}_+. \end{equation} \tag{ 1 }$$

The non-negative parameter α is called degree of connectivity: if the sites i are regarded as vertices of a graph, and the pairs (i_ν, j_ν) define the edges of this graph, then each vertex is the endpoint of on average 2α edges as explained below.

The Hamiltonian (1) as written has the advantage that it is the sum of (a random number of) i.i.d. terms. To see the connection to the original VB Hamiltonian, note that the Poisson distributed total number of bonds obey $P_{\alpha N}=\alpha N + O(\sqrt{N})$ for large N. As there are N² ordered spin pairs (i, j), each obtains a bond with the probability ∼α/N for large N. The probabilities of obtaining two, three (and so on) bonds scale as 1/N², 1/N³, ... so can be neglected. The probability of having a bond between any unordered pair of spins is twice as large, i.e. 2α/N. For large N, each site therefore has on average 2α bonds connecting to it, and more precisely, this number of bonds to each site has a Poisson distribution with mean 2α. The self-loops that we have allowed just add σ-independent constant to the Hamiltonian so are irrelevant.

We will denote by $\mathbb {E}$ the expectation with respect to all the (quenched) variables, i.e. all the random variables except the spins, collectively denoted by $\mathcal {J}$. The Gibbs measure ω is defined by

$$\begin{equation*} \omega (\varphi )=\frac{1}{Z_N(\alpha ,\beta )}\sum _{\sigma } \varphi (\sigma ) \,{\rm e}^{-\beta H_N(\sigma , \alpha ; \mathcal {J})} \end{equation*}$$

for any observable $\varphi :\lbrace -1,+1\rbrace ^{N}\rightarrow \mathbb {R}$, where $Z_N(\alpha ,\beta )= \sum _{\sigma }\exp (-\beta H_{N}(\sigma ,\alpha ;\mathcal {J}))$ is the partition function for a finite number of spins N. When dealing with more than one configuration, the product Gibbs measure will be denoted by Ω, and spin configurations taken from each space in such a product are called 'replicas'. We use the symbol 〈.〉 to mean $\langle . \rangle = \mathbb {E}\Omega (.)$.

We will often omit the dependence on the size of the system N of various quantities, a convention already deployed above. In general, we will allow slight abuses of notation to lighten the expressions as long as there is no risk of confusion. The pressure P_N(α, β) and the free energy density f_N(α, β) for a given system size N are defined by

$$\begin{equation*} P_N(\alpha ,\beta )= -\beta f_{N}(\alpha ,\beta )=\frac{1}{N}\mathbb {E}\ln Z_{N}(\alpha ,\beta ), \end{equation*}$$

and we assume that the limit lim_{N → ∞}P_N(α, β) = P(α, β) exists.

The entire physical behavior of the model is encoded by the distribution of the (even) multi-overlaps q_1...2n, which are functions of several configurations σ⁽¹⁾, σ⁽²⁾, ..., σ⁽²ⁿ⁾ defined by

$$\begin{equation*} q_{1\ldots 2n}=\frac{1}{N} \sum _{i=1}^N \sigma _i^{(1)}\cdots \sigma _i^{(2n)}. \end{equation*}$$

By studying the behavior of these order parameters, it is possible to obtain a phase diagram for diluted spin glasses in the (α, β) plane which consists of an ergodic phase (where all overlaps vanish in the thermodynamic limit of large N) and a spin glass phase (where the overlaps are positive), separated by a second-order critical line given by

$$\begin{equation} 2\alpha \tanh (\beta )=1. \end{equation} \tag{ 2 }$$

Following the idea at the heart of the cavity approach (namely measuring the effect on the free energy of the addition of one spin to the system; see [3, 29] for a summary), we write, in distribution,

$$\begin{equation} H_{N+1}(\sigma , \sigma _{N+1}, \alpha ; \mathcal {J}) = -\sum _{\nu =1}^{P_{\alpha \frac{N^{2}}{N+1}}} J_{\nu }\sigma _{i_{\nu }}\sigma _{j_{\nu }} -\sum _{\nu =1}^{P_{\alpha \frac{2N}{N+1}}} J^{\prime }_{\nu }\sigma _{i^{\prime }_{\nu }}\sigma _{N+1} -\sum _{\nu =1}^{P_{\frac{\alpha }{N+1}}} J^{\prime \prime }_{\nu }\sigma _{N+1}^2, \end{equation} \tag{ 3 }$$

where σ_{N + 1} is the added spin. {J'_ν, J''_ν} are independent copies of J, and {i_ν}, {j_ν} and {i'_ν} are independent random variables all uniformly distributed over {1, ..., N}. The last term in (3) does not contribute when N is large, and at any rate is a constant which cancels from the Boltzmann measure.

Note that we can equivalently write the above decomposition as

$$\begin{equation} H_{N+1}(\sigma , \sigma _{N+1}, \alpha ; \mathcal {J}) = H_{N}(\sigma ,\alpha ^{\prime };\mathcal {J}) +h_{N+1}\sigma _{N+1}, \end{equation} \tag{ 4 }$$

where

$$\begin{equation*} \alpha ^{\prime }=\alpha \frac{N}{N+1} ,\quad h_{N+1}=-\sum _{\nu =1}^{P_{2 \alpha ^{\prime }}}J^{\prime }_{\nu }\sigma _{i^{\prime }_{\nu }}. \end{equation*}$$

Exploiting the additivity property of Poisson variables, we can also decompose the Hamiltonian for an N-spin system so that it shares the first term with H_{N + 1}:

$$\begin{equation} H_{N}(\sigma ,\alpha ;\mathcal {J})=H_{N}(\sigma ,\alpha ^{\prime }; \mathcal {J}) +H_{N}(\sigma ,\alpha ^{\prime }/N; {\hat{\mathcal J}}), \end{equation} \tag{ 5 }$$

where the two Hamiltonians on the right-hand side (rhs) have independent quenched random variables $\mathcal {J}$ and ${\hat{\mathcal J}}$. Hence, if we call

$$\begin{equation*} H_{N}(\sigma ;\alpha ^{\prime }/N; {\hat{\mathcal J}}) =\hat{H}_N(\sigma ,\alpha ^{\prime };{\hat{\mathcal J}})= -\sum _{\nu =1}^{P_{\alpha ^{\prime }}}\hat{J}_{\nu } \sigma _{\hat{i}_{\nu }}\sigma _{\hat{j}_{\nu }}, \end{equation*}$$

then

$$\begin{equation*} \mathbb {E}\ln \frac{Z_{N+1}(\alpha ,\beta )}{Z_{N}(\alpha ,\beta )} =\mathbb {E}\ln \frac{\sum _{\sigma ,\sigma _{N+1}} \xi _{\sigma }\exp (-\beta h_{N+1}\sigma _{N+1})}{\sum _{\sigma } \xi _{\sigma }\exp (-\beta \hat{H}(\sigma ,\alpha ^{\prime };{\hat{\mathcal J}}))}, \end{equation*}$$

with

$$\begin{equation*} \xi _{\sigma }=\exp (-\beta H_N(\sigma ,\alpha ^{\prime };\mathcal {J})). \end{equation*}$$

As elegantly explained in [3], and discussed in detail in [29], this equation expresses the incremental contribution to the free energy in terms of the mean free energy of a spin added to a reservoir whose internal state is described by (σ, ξ_σ), corrected by an inverse-fugacity term $\hat{H}$, which encodes a connectivity shift. The former may be thought of as the cavity into which the (N + 1) particle is added: for N ≫ 1, the value of the added spin, σ_{N + 1}, does not significantly affect the field that would act for the next increment in N. Hence, for the next addition of a particle, we may continue to regard the state of the reservoir as given by just the configuration σ. However, the weight of the configuration (which is still to be normalized to yield the probability of the configuration) changes according to

$$\begin{equation*} \xi _{\sigma }\rightarrow \xi _{\sigma }\,{\rm e}^{-\beta h_{N+1}\sigma _{N+1}}. \end{equation*}$$

This transformation is called the cavity technique.

The addition of a new spin can, because of the randomness of the couplings, effectively be regarded as an external random field that vanishes in the thermodynamic limit. This is essentially the perspective of the stochastic stability approach [8, 9].

By an interpolation method [4], the (N + 1)th spin can be added to the N-spin system smoothly via an appropriately defined cavity function Ψ(α, β, t), t ∈ [0, 1], which reads

$$\begin{equation} \Psi (\alpha ,\beta ,t) =\mathbb {E}\ln \omega (\,{\rm e}^{\beta \sum _{\nu =1}^{P_{2\alpha t}} J^{\prime }_{\nu }\sigma _{i^{\prime }_{\nu }}}). \end{equation} \tag{ 6 }$$

Due to the gauge symmetry of the VB model, namely the symmetry $\sigma _{i_{\nu }}\rightarrow \sigma _{i_{\nu }}\sigma _{N+1}$ (whose action leaves the VB Hamiltonian invariant), the above cavity function turns out to contain an effective two-body interaction, as in the original Hamiltonian, and the sum over σ_{N + 1} = ±1 in the partition function gives a trivial factor 2 because σ_{N + 1} plays the role of a hidden variable; this factor 2 yields, once the logarithm is taken, just the high temperature entropy.

Inspired by the cavity perspective, taking φ as a generic function of the spin configuration, we can define a generalized Boltzmann measure (denoted by the subscript 〈.〉_t) as

$$\begin{equation*} \omega _t(\varphi )=\frac{\omega (\varphi (\sigma ) \,{\rm e}^{\beta \sum _{\nu =1}^{P_{2\alpha t}}J^{\prime }_{\nu }\sigma _{i^{\prime }_{\nu }}})}{\omega ({\rm e}^{\beta \sum _{\nu =1}^{P_{2\alpha t}}J^{\prime }_{\nu }\sigma _{i^{\prime }_{\nu }}})}. \end{equation*}$$

Note that in the t = 0 case, we always recover the unperturbed Boltzmann measure of an N-spin system and in the t = 1 case, we recover the unperturbed Boltzmann measure of an (N + 1)-spin system, with a small shift in the connectivity that becomes negligible in the thermodynamic limit.

Let us now briefly describe the stochastic stability properties for averaged overlap correlation functions (OCFs); these will become useful shortly. We split OCFs into two categories: filled OCFs, showing shape robustness with respect to the stochastic perturbation, and fillable OCFs, showing saturability with respect to the same perturbation.

• Filled OCFs are monomials in overlaps among s replicas such that each replica appears an even number of times. Examples are q²₁₂, q²₁₂₃₄ or q₁₂q₂₃q₁₃.
• Fillable OCFs are overlap monomials among s replicas which become filled when multiplied by a single overlap among exactly those replicas appearing only an odd number of times. Examples are q₁₂, q₁₂₃₄ or q₁₂q₁₃.

It should be pointed out that all monomial OCFs are either filled or fillable, because one can always find a multi-overlap to fill any (monomial) OCF that is not filled. This contrasts with the case of the SK model, where only overlaps among two replicas can be used to fill an OCF [4]. The division into filled and fillable OCFs is made because of differences in how their averages react to the perturbing field induced by the cavity function [4, 6, 29]. In the thermodynamic limit, the averages of the filled OCFs become independent of t, i.e.

$$\begin{equation*} \lim _{N \rightarrow \infty }\partial _t \langle \mbox{filled OCF}\rangle _t = 0. \end{equation*}$$

We refer to this property as robustness.

On the other hand, using the gauge symmetry, one has in the thermodynamic limit that the averages of fillable OCFs become filled at t = 1, namely [4, 6, 29]

$$\begin{equation*} \lim _{N \rightarrow \infty } \langle \mbox{fillable OCF} \rangle _{t=1} = \lim _{N \rightarrow \infty } \langle \mbox{filled OCF} \rangle _{t=1}= \lim _{N \rightarrow \infty } \langle \mbox{filled OCF} \rangle . \end{equation*}$$

We refer to this last property as saturability. Note that we have dropped the subscript t in the last equality because of the robustness of filled OCFs. Examples of saturability are 〈q₁₂〉_{t = 1} = 〈q²₁₂〉, 〈q₁₂₃₄〉_{t = 1} = 〈q²₁₂₃₄〉 and 〈q₁₂q₁₃〉_{t = 1} = 〈q₁₂q₁₃q₂₃〉, with the limit N → ∞ always understood.

We only sketch the proof of the above propositions and refer the reader to [4–6, 29] for a detailed discussion and proofs. Let us show how the fillable OCFs turn out to become filled OCFs in the N → ∞ limit at t = 1. The stability of the filled OCFs will then be a straightforward consequence of their gauge invariance, which is heavily used in the proof. Consider the simplest case of a monomial Q_ab that is fillable by multiplying by q_ab, with replicas a and b each appearing only once in Q_ab. Then we can write

$$\begin{equation*} \langle Q_{ab} \rangle _t = \left\langle \sum _{ij}\big(\sigma _i^a\sigma _j^b\big/N^2\big)Q_{ij}(\sigma )\right\rangle _t, \end{equation*}$$

where Q_ij contains all factors that do not depend on replicas a or b. Factorizing the state Ω_t, we obtain

$$\begin{eqnarray*} &&\langle Q_{ab} \rangle _t = \frac{1}{N^2}\mathbb {E}\left( \sum _{ij} \omega _t(\sigma _i^a)\omega _t(\sigma _j^b)\Omega _t(Q_{ij}) \right). \end{eqnarray*}$$

Now rewrite the last expression for t = 1: by applying the gauge transformation σ_i → σ_iσ_{N + 1}, the states acting on the replicas a and b are ω_{t = 1}(σ^a_i) → ω(σ^a_iσ_{N + 1}^a) + O(N⁻¹), while the remaining product state Ω_t continues to act on an even number of occurrences of each replica and is not modified (in a manner directly analogous to the robustness of averages of filled OCFs). Putting all the replicas back into a single product state, we have

$$\begin{equation} \omega \big(\sigma _i^a\sigma _{N+1}^a\big)\omega \big(\sigma _i^b\sigma _{N+1}^b\big)\Omega (Q_{ij}) = \Omega \big(\sigma _i^a\sigma _j^b\sigma _{N+1}^a\sigma _{N+1}^bQ_{ij}\big). \end{equation} \tag{ 7 }$$

Now the index N + 1 can be replaced by a dummy index k that is averaged according to 1 = N⁻¹∑^N_{k = 1}; this does not change the result except for O(N⁻¹) corrections—from values of k that coincide with i, j, or further summation indices in Q_ij—that vanish in the thermodynamic limit. Since N⁻¹∑^N_{k = 1}σ_k^aσ^b_k = q_ab, this gives the desired result.

In this subsection we want to show that the free energy density can be written in terms of an 'energy-like' contribution and an 'entropy-like' one. As a consequence of this decomposition, and given that we have previously investigated the constraints deriving from the energy-like term, we will then restrict our investigation to the entropy-like contribution, which (as we are going to show) is encoded in the cavity function.

It is in fact always possible, via the fundamental theorem of calculus, to relate the free energy to its derivative with respect to a chosen parameter, here the connectivity α. Clearly, the result is a relation between the free energy and its α-derivative where, interestingly, the missing term is exactly the cavity function. In the thermodynamic limit, this decomposition takes the form

$$\begin{equation} P(\alpha ,\beta ) + \alpha \partial _\alpha P(\alpha ,\beta ) = \ln 2 + \Psi (\alpha ,\beta ,t=1). \end{equation} \tag{ 8 }$$

We emphasize that the equation above, which we are going to prove using continuity and the fundamental theorem of calculus, can be thought of as a generalized thermodynamic definition of the free energy. In this approach, the cavity function naturally acts as the thermodynamic entropy, which is why its investigation suggested the title of this paper.

To see briefly how (8) arises, let us write down the partition function of a system of N + 1 spins at connectivity α* = α(N + 1)/N, using the decomposition (3) of the relevant Hamiltonian:

$$\begin{eqnarray*} Z_{N+1}(\alpha ^*,\beta ) &=& {\rm e}^{\beta \sum _{\nu =1}^{P_{\alpha /N}} J^{\prime \prime }_\nu } \sum _{\sigma , \sigma _{N+1}= \pm 1} \,{\rm e}^{\beta \sum _{\nu =1}^{P_{\alpha N}}J_{\nu }\sigma _{i_\nu }\sigma _{j_{\nu }} +\beta \sum _{\nu =1}^{P_{2\alpha }} J^{\prime }_\nu \sigma _{i^{\prime }_\nu }\sigma _{N+1}} \\ &=& {\rm e}^{\beta \sum _{\nu =1}^{P_{\alpha /N}} J^{\prime \prime }_\nu } \sum _{\sigma , \sigma _{N+1}= \pm 1} \,{\rm e}^{\beta \sum _{\nu =1}^{P_{\alpha N}}J_{\nu }\sigma _{i_\nu }\sigma _{j_{\nu }} +\beta \sum _{\nu =1}^{P_{2\alpha }} J^{\prime }_\nu \sigma _{i^{\prime }_\nu }} \\ &=&2\,{\rm e}^{\beta \sum _{\nu =1}^{P_{\alpha /N}} J^{\prime \prime }_\nu } \sum _{\sigma } \,{\rm e}^{-\beta H_N(\sigma ,\alpha ;\mathcal {J}) +\beta \sum _{\nu =1}^{P_{2\alpha }} J^{\prime }_\nu \sigma _{i^{\prime }_\nu }}, \end{eqnarray*}$$

where in going from the first to the second line, we have gauge transformed σ_i → σ_iσ_{N + 1}. Multiplying and dividing by Z_N(α, β) and taking logs, we obtain

$$\begin{eqnarray*} &&\ln Z_{N+1}(\alpha ^*,\beta )= \ln 2 + \beta \sum _{\nu =1}^{P_{\alpha /N}} J^{\prime \prime }_\nu +\ln Z_N(\alpha ,\beta ) +\ln \omega \big({\rm e}^{\beta \sum _{\nu =1}^{P_{2\alpha }} J^{\prime }_\nu \sigma _{i^{\prime }_{\nu }}}\big). \nonumber \end{eqnarray*}$$

Averaging over the disorder removes the second term and transforms the last one into the cavity function at t = 1. Rearranging slightly, the result reads

$$\begin{eqnarray*} &&\fl [\mathbb {E}\ln Z_{N+1}(\alpha ^*,\beta )-\mathbb {E}\ln Z_{N+1}(\alpha ,\beta )]+ [\mathbb {E}\ln Z_{N+1}(\alpha ,\beta )\, -\,\mathbb {E}\ln Z_N (\alpha ,\beta )] \nonumber\\ &&=\ln 2 + \Psi _N(\alpha ,\beta ,t=1). \end{eqnarray*}$$

Now α* − α = α/N becomes small as N grows so we can Taylor expand the first difference on the left-hand side (lhs) as $(\alpha /N)\partial _\alpha \mathbb {E} \ln Z_{N+1}(\alpha ,\beta )+O(1/N)$. The second difference on the lhs, on the other hand, gives the pressure as N → ∞, and so the decomposition (8) follows in the limit.

To go beyond the heuristic arguments reviewed in the previous section, we need to write down first the general form of the streaming equation (10). The t-dependence in the Boltzmann factor (9) arises only from the Poisson variable $P_{2\alpha ^{\prime }t}$. For a generic function of such a variable, leaving off the factor 2α' initially, one has $\mathbb {E}f(P_t) = {\rm e}^{-t}\sum _{k=0}^\infty f(k) t^k/k!$ and so

$$\begin{equation*} \fl \partial _t \mathbb {E}f(P_t) = -{\rm e}^{-t}\sum _{k=0}^\infty f(k) t^k/k! + {\rm e}^{-t} \sum _{k=1}^\infty f(k) t^{k-1}/(k-1)! = -\mathbb {E}f(P_t) + \mathbb {E}f(1+P_t). \end{equation*}$$

Applying this to $\langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}$ gives

$$\begin{equation*} \frac{\partial _t \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{2\alpha ^{\prime }} = - \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t} + \mathbb {E} \frac{\Omega _{\alpha ^{\prime },\beta ^{\prime },t}(F_s \,{\rm e}^{\beta ^{\prime } J \sum _{a=1}^s\sigma _i^a})}{\Omega _{\alpha ^{\prime },\beta ^{\prime },t}({\rm e}^{\beta ^{\prime } J \sum _{a=1}^s\sigma _i^a})}, \end{equation*}$$

where J is the random interaction strength for the additional coupling and i, drawn uniformly from {1, ..., N}, is the associated spin index. The resulting exponential can be written as ∏^s_{a = 1}[cosh (β') + sinh (β')Jσ_i^a] and canceling the common factor [cosh (β')]^s yields

$$\begin{equation*} \frac{\partial _t \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{2\alpha ^{\prime }} + \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t} = \mathbb {E} \frac{\Omega _{\alpha ^{\prime },\beta ^{\prime },t}\big(F_s \prod _{a=1}^s[1+J\theta \sigma _i^a]\big)}{\Omega _{\alpha ^{\prime },\beta ^{\prime },t}\big(\prod _{a=1}^s[1+J\theta \sigma _i^a]\big)}, \end{equation*}$$

where θ = tanh (β') as before. The denominator factorizes across replicas, giving a factor $[1+J\theta \Omega _{\alpha ^{\prime },\beta ^{\prime },t}(\sigma _i)]^{-s}$. One expands this and also the numerator in powers of θ to obtain for the rhs of the last equation

$$\begin{equation*} \fl \mathbb {E} \Omega _{\alpha ^{\prime },\beta ^{\prime },t}\left(F_s \sum _{k=0}^s(J\theta )^k \sum _{1\le a_1<\cdots <a_k\le s} \sigma _i^{a_1} \cdots \sigma _i^{a_k}\right) \sum _{l=0}^\infty \frac{(s+l-1)!}{l!(s-1)!}(-1)^l (J\theta )^l [\Omega _{\alpha ^{\prime },\beta ^{\prime },t}(\sigma _i)]^l. \end{equation*}$$

To cast each term as an average over a replicated measure again, one can write $[\Omega _{\alpha ^{\prime },\beta ^{\prime },t}(\sigma _i)]^l = \Omega _{\alpha ^{\prime },\beta ^{\prime },t}(\sigma _i^{s+1}\cdots \sigma _i^{s+l})$. This gives as the general streaming equation

$$\begin{eqnarray*} &&\fl \frac{\partial _t \langle F_s\rangle }{2\alpha ^{\prime }} = \left\langle F_s\left[-1+ \sum _{k=0}^s(J\theta )^k \sum _{1\le a_1<\cdots <a_k\le s} \sigma _i^{a_1} \cdots \sigma _i^{a_k}\right.\right.\nonumber\\ &&\left.\left.\times \sum _{l=0}^\infty \frac{(s+l-1)!}{l!(s-1)!}(-1)^l (J\theta )^l \sigma _i^{s+1}\cdots \sigma _i^{s+l}\right]\right\rangle , \end{eqnarray*}$$

where for brevity we have dropped the subscript α', β', t on all averages. Now we gather terms according to equal powers m = k + l of θ, and note that the m = 0 term cancels with −1:

$$\begin{equation*} \fl \frac{\partial _t \langle F_s\rangle }{2\alpha ^{\prime }} = \left\langle F_s\sum _{m=1}^\infty (J\theta )^{m} \sum _{l=0}^{m} \sum _{1\le a_1<\cdots <a_{m-l}\le s} \sigma _i^{a_1} \cdots \sigma _i^{a_{m-l}} \frac{(s+l-1)!}{l!(s-1)!}(-1)^{l} \sigma _i^{s+1}\cdots \sigma _i^{s+l}\right\rangle . \end{equation*}$$

Here and below it is understood that the sum over a₁, ..., a_{m − l} vanishes when m − l > s, because it is then not possible to satisfy the constraint 1 ⩽ a₁ < ⋅⋅⋅ < a_{m − l} ⩽ s.

Performing now the expectation over J cancels all odd orders m, and the expectation over i produces a multi-overlap, giving

$$\begin{equation*} \fl \frac{\partial _t \langle F_s\rangle }{2\alpha ^{\prime }} = \left\langle F_s\sum _{m=2,4,\ldots } \theta ^{m} \sum _{l=0}^{m} \frac{(s+l-1)!}{l!(s-1)!}(-1)^{l} \sum _{1\le a_1<\cdots <a_{m-l}\le s} q_{a_1\ldots a_{m-l},s+1\ldots s+l}\right\rangle . \end{equation*}$$

To state this result in a compact form, we define

$$\begin{equation*} \nonumber C^{(m,n)}_s = \sum _{l=0}^{m} \frac{(s+l-1)!}{l!(s-1)!}(-1)^{l} \sum _{1\le a_1<\cdots <a_{m-l}\le s} q^n_{a_1\ldots a_{m-l},s+1\ldots s+l}. \end{equation*}$$

The superscript m indicates how many replicas are involved in each of the overlaps in this expression. Each overlap is taken to the power n, a generalization which will be useful shortly. Then after re-instating the subscripts on the averages, the streaming equation can be written simply as

$$\begin{equation} \frac{\partial _t \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{2\alpha ^{\prime }} = \left\langle F_s\sum _{m=2,4,\ldots } \theta ^{m} C_s^{(m,1)}\right\rangle _{\alpha ^{\prime },\beta ^{\prime },t}. \end{equation} \tag{ 11 }$$

We can now state the arguments of the previous sections in a more general form. If F_s is filled, then the derivative on the lhs of (11) must vanish. If one evaluates at α' = α, β' = β, t = 1 and for N → ∞, all the overlaps in C^(m)_s become squared—the fillable averages become filled—so each factor C^{(m, 1)}_s turns into a 'higher order AC factor' C^{(m, 2)}_s. The identities from directed stochastic stability are therefore (for filled F_s and in the limit N → ∞)

$$\begin{equation*} \nonumber \left\langle F_s\sum _{\gamma _{\rm o}=2,4,\ldots } \theta ^{m} C_s^{(m,2)}\right\rangle = 0. \end{equation*}$$

As explained above, one cannot necessarily separate the different orders in θ in this result. This is possible by 'reversing' the approach, as we will now see.

Progress in demonstrating that each term of the above expression must vanish separately, i.e. 〈F_sC^{(m, 2)}_s〉 = 0 for each s and even m, can be made by considering the derivatives not at t = 1 but at t = 0 and assuming that, in general, stochastic stability holds [8, 9]. Let us take a generic overlap polynomial F_s, and consider as above the 'smooth cavity' perturbation with a modified temperature β' and corresponding θ = tanh (β'). Then the first derivative w.r.t. t is (writing m = 2n now)

$$\begin{equation*} \nonumber \frac{\partial _t \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{2\alpha ^{\prime }} = \sum _{n=1}^\infty \theta {}^{2n} \left\langle F_s C_s^{(2n,1)}\right\rangle _{\alpha ^{\prime },\beta ^{\prime },t}. \end{equation*}$$

Pulling the infinite sum out of the expectation and differentiating again (assuming that we can interchange differentiation with the infinite sum over n), we obtain

$$\begin{equation} \frac{\partial ^2_t \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{(2\alpha ^{\prime })^2} = \sum _{n=1}^\infty \sum _{m=1}^\infty \theta ^{2(m+n)} \big\langle F_s C_s^{(2n,1)}C_{s+2n}^{(2m,1)}\big\rangle _{\alpha ^{\prime },\beta ^{\prime },t}. \end{equation} \tag{ 12 }$$

From now on, take F_s to be filled and set t = 0 instead of t = 1: we can then drop the subscripts on the average on the rhs because at t = 0, we have an unperturbed Boltzmann average. Then because of gauge invariance of this unperturbed state, all terms with m ≠ n give vanishing averages: F_s is filled, C^{(2n, 1)}_s consists of a sum of nth-order overlaps of n distinct replicas and C^{(2m, 1)}_{s + 2n} of a sum of mth-order overlaps of m distinct replicas. At least |m − n| ⩾ 1 replicas therefore occur an odd number of times in all possible combinations of terms. Hence we can collapse the sum to

$$\begin{equation} \left.\frac{\partial _t^2 \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{(2\alpha ^{\prime })^2}\right|_{t=0} = \sum _{n=1}^\infty \theta ^{2n} \big\langle F_s C_s^{(2n,1)}C_{s+2n}^{(2n,1)}\big\rangle . \end{equation} \tag{ 13 }$$

One can simplify further: C^{(2n, 1)}_s is a function only of replicas 1, ..., s + 2n. This means that in the sum (11) defining C^{(2n, 1)}_{s + 2n}, only the term with l = 0 can give non-vanishing averages, because all other terms depend on some of the replicas s + 2n + 1, ..., s + 4n and these would remain unpaired. In other words,

$$\begin{equation*} \big\langle F_s C_s^{(2n,1)}C_{s+2n}^{(2n,1)}\big\rangle = \left\langle F_s C_s^{(2n,1)} \sum _{1\le b_1<\cdots <b_{2n} \le s+2n} q_{b_1\ldots b_{2n}} \right\rangle . \end{equation*}$$

But now for each term $q_{a_1,\ldots ,a_{2n}}$ in C^{(2n, 1)}_s (with 1 ⩽ a₁ < ⋅⋅⋅ < a_2n ⩽ s + 2n), there is exactly one entry in the sum over 1 ⩽ b₁ < ⋅⋅⋅ < b_2n ⩽ s + 2n which fills this term and gives a nonzero average, namely b₁ = a₁, ..., b_2n = a_2n. The multiplication by the sum over b₁, ..., b_2n therefore just has the effect of squaring all the overlaps in C^{(2n, 1)}_s and we obtain

$$\begin{equation*} \big\langle F_s C_s^{(2n,1)}C_{s+2n}^{(2n,1)}\big\rangle = \big\langle F_s C_s^{(2n,2)}\big\rangle . \end{equation*}$$

Inserting into (13) then gives

$$\begin{equation*} \nonumber \left.\frac{\partial _t^2 \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{(2\alpha ^{\prime })^2}\right|_{t=0} = \sum _{n=1}^\infty \theta {}^{2n} \left\langle F_s C_s^{(2n,2)}\right\rangle . \end{equation*}$$

Now we assume that we have stochastic stability for generic θ [8, 9] and t, i.e. that $\langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}$ is independent of β' and t in the limit N → ∞. Then the last expression vanishes and hence so must all coefficients of different powers of θ. We thus obtain the relations (for s ⩾ 2 and n ⩾ 1)

$$\begin{equation} \lim _{N\rightarrow \infty } \langle F_s C_s^{(2n,2)}\rangle = 0. \end{equation} \tag{ 14 }$$

This includes for n = 1 all the standard SK AC identities, but also all their higher order generalizations.

We now compare closely related identities obtained by De Sanctis and Franz [15] and Franz et al [16]. It is easy to check that the identities obtained in these papers can be written in our notation as

$$\begin{equation*} \lim _{N\rightarrow \infty } \big\langle q_{1\ldots s}^{2r} C_s^{(2n,2p)}\big\rangle = 0, \end{equation*}$$

for arbitrary positive integers r and p. The identities (14) relate to p = 1 but are then rather more general because they allow arbitrary filled overlap polynomials for F_s. For example, our identities for F₃ = q₁₂q₂₃q₁₃ are not contained in the set of identities from [15, 16].

The generalization of the identities (14) to exponents greater than 2 can be achieved relatively simply by a standard approach, allowing not just fields but p-spin interactions in the perturbing term. The perturbation term from (9) would then be generalized to

$$\begin{equation*} \beta ^{\prime } \sum _{\nu =1}^{P_{2\alpha ^{\prime } t}} \tilde{J^{\prime }}_{\nu }\sigma _{i^1_{\nu }} \cdots \sigma _{i^{p-1}_{\nu }}, \end{equation*}$$

and reduces to the latter for p = 2 as it should.

In the streaming equation w.r.t. t, factors like σ^a_i are then consistently replaced by $\sigma _{i^1}^a \cdots \sigma _{i^{p-1}}^a$, and accordingly one obtains in the end

$$\begin{equation} \frac{\partial _t \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{2\alpha ^{\prime }} = \left\langle F_s\sum _{m=2,4,\ldots } \theta ^{m} C_s^{(m,p-1)}\right\rangle _{\alpha ^{\prime },\beta ^{\prime },t}. \end{equation} \tag{ 15 }$$

The calculation of the second derivative at t = 0 generalizes in the same way, giving

$$\begin{equation} \left.\frac{\partial _t^2 \langle F_s\rangle _{\alpha ^{\prime },\beta ^{\prime },t}}{(2\alpha ^{\prime })^2}\right|_{t=0} = \sum _{n=1}^\infty \theta {}^{2n} \left\langle F_s C_s^{(2n,2(p-1))}\right\rangle . \end{equation} \tag{ 16 }$$

Based again on the assumption of stochastic stability under a small perturbation of the Boltzmann measure caused by the introduction of O(N⁰) random (p − 1)-spin interaction terms into a system of N spins, one then deduces the identities

$$\begin{equation} \lim _{N\rightarrow \infty } \big\langle F_s C_s^{(2n,2(p-1))}\big\rangle = 0. \end{equation} \tag{ 17 }$$

With the overlap exponent now generalized from 2 to 2(p − 1), these form a strict superset of the identities from [16, 15].

We have been somewhat casual above in not distinguishing between odd and even orders p − 1 of the perturbing Hamiltonian. Specifically, the reasoning that leads to the simplified form of the second derivative (16) works, by analogy with the case p = 2, only when p − 1 is odd. For even p − 1, a more complicated expression analogous to (12) would result. However, in this case one can exploit the vanishing of the first derivative (15). Writing p − 1 = 2(p' − 1) because p − 1 is even, one then obtains again the identities (17), with p replaced by p' which is now an arbitrary integer.

While we have assumed throughout Poissonian graphs, where each pair interaction is present independently of the others, one would expect that the identities (17) hold also for a spin glass model on other graphs with finite connectivity. The treatment in [16] is more general in this regard, and it may be possible to adapt the methods used there to generalize (17) to this broader range of settings.

Our reasoning leading to the general higher order AC identities (17) is, like the one in [16, 15], not rigorous. The main assumption here, as in other papers, is stochastic stability for general values of β' ≠ β. More technically, we have also been somewhat cavalier in our treatment of infinite sums, interchanging e.g. differentiation w.r.t. t with summation.