The hierarchical cavity formula for diluted spin glasses

Source and verification

This article explains family 221 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A diluted spin glass has only finitely many interactions per spin on average. Its cavity description therefore uses random local messages rather than one deterministic effective field. Family 221 claims the exact Mézard–Parisi variational formula for a specified even-arity interaction class, using an infimum over finite hierarchies.

🏷️ Preliminaries: a sparse interaction model

Take $N$ spins ${\sigma }_i\in \{-1,1\}$ and an independent number $M_N\sim \mathrm{Poisson}(\alpha N)$ of interactions. Each interaction chooses $p$ uniform vertex indices independently, with replacement, where $p\ge 2$ is even. If ${\theta }_k$ are iid random interaction functions and $h_i$ iid fields, the partition function is

$$\begin{array}{rl}{\mathcal{H}}_N(\sigma )& =\sum _{k=1}^{M_N}{\theta }_k({\sigma }_{i_{1,k}},\dots ,{\sigma }_{i_{p,k}})\\ & +\sum _i{h}_i{\sigma }_i,\\ Z_N& =\sum _{\sigma }{e}^{{\mathcal{H}}_N(\sigma )}.\end{array}$$

All disorder families are mutually independent. The pressure is $F_N=N^{-1}\mathbb{E}\log Z_N$; the physical free energy includes an additional negative inverse-temperature factor.

A cavity message $x\in \mathbb{R}$ is a one-spin probability law

$$q_x(s)=\frac{e^{xs}}{2\cosh x},s\in \{-1,1\}.$$

It represents the effective field felt when a neighboring interaction is removed. At low temperature, a distribution of messages may itself depend on which competing Gibbs state is sampled. A hierarchy records successive distributions of those distributions.

🏷️ The interaction assumptions are part of the theorem

The manuscript assumes a factorization

$$e^{\theta (s_1,\dots ,s_p)}=a\left(1+b\prod _{j=1}^p{f}_j(s_j)\right),$$

where $a>0$, the $f_j$ are independent copies of one random function, $b$ is independent of them, the absolute value of the product term is strictly below one, and $\mathbb{E}(-b{)}^k\ge 0$ for every positive integer $k$. It also assumes only the first moments

$$\mathbb{E}\underset{s}{\max }|\theta (s)|+\mathbb{E}|h|<\infty .$$

These are the Panchenko–Talagrand class assumptions. “Diluted spin glass” by itself is not enough to guarantee them. For example, the familiar symmetric two-spin interaction $\theta (s,t)=Jst$ has $a=\cosh J$, $b=\tanh J$, and $f(s)=s$; symmetry makes odd moments of $b$ vanish and even moments nonnegative.

🏷️ The finite-hierarchy formula

Given messages, define the interaction average $C_{\theta }$ and the message sent to a new spin $\epsilon$ by

$$\begin{array}{rl}& C_{\theta }(x_1,\dots ,x_p)\\ & =\sum _{s_1,\dots ,s_p}{e}^{\theta (s)}\prod _j{q}_{x_j}(s_j),\\ & U_{\theta }(x_1,\dots ,x_{p-1};\epsilon )\\ & =\log \sum _{s_1,\dots ,s_{p-1}}{e}^{\theta (s,\epsilon )}\prod _{j<p}{q}_{x_j}(s_j).\end{array}$$

Let $K\sim \mathrm{Poisson}(\alpha p)$ be the new site’s interaction count. With independent message labels, set

$$\begin{array}{rl}A(\epsilon )& =h\epsilon +\sum _{j=1}^K{U}_{{\theta }_j}(x^j;\epsilon ),\\ V_s& =\frac{1}{2}\sum _{\epsilon =\pm 1}{e}^{A(\epsilon )},\\ V_e& =C_{\theta }(x_1,\dots ,x_p).\end{array}$$

At depth $r$, successive random message laws are sampled along a finite hierarchy. Increasing numbers $0<m_1<\cdots <m_r<1$ specify conditional power means

$$\begin{array}{c}{T}_{r}V=V,\\ (T_{d}V{)}^{m_{d+1}}=\mathbb{E}[(T_{d+1}V{)}^{m_{d+1}}\mid {\mathcal{F}}_d].\end{array}$$

Here ${\mathcal{F}}_d$ retains the physical disorder and message laws revealed through level $d$; only the remaining auxiliary message randomness is integrated. Different message labels start with independent draws from the trial hierarchy law. At depth zero, use iid messages and omit these operators.

Claimed variational equality

The limit of $F_N$ equals the infimum, over every finite depth and every admissible trial hierarchy and mass sequence, of

$$\begin{array}{rl}& \log 2+\mathbb{E}\log T_0{V}_s\\ & -\alpha (p-1)\mathbb{E}\log T_0{V}_e.\end{array}$$

No minimizing hierarchy or infinite-depth Gibbs-state construction is assumed to exist. (OpenAI, 2026)

The interaction correction avoids overcounting interactions incident to the new site. When interactions and fields vanish, $V_s=V_e=1$ and the formula returns $\log 2$, as required by $Z_N=2^N$.

🏷️ The main obstacle: all replica patterns

An interpolation gives the upper bound for each trial hierarchy. The proposed matching lower bound must recover independent cavity messages from a common finite system. Pair overlaps alone do not determine the joint pattern of several replicas; the argument needs all finite multioverlaps, spatial averages of products of replica spins.

Small bounded Poisson perturbations select systems with concentrated insertion scores. A finite-tree expansion prescribes replica branching patterns. Delaying branch levels by one step yields an induction controlling multioverlaps averaged over branching depths. This averaged control is enough because the cavity expansion itself averages over such trees.

Only after this control is established can message roots at distinct sites be replaced by independent draws. The system size first tends to infinity at fixed depth, and depth increases afterward. Uniform truncation estimates finally remove boundedness of the disorder using its first moments. This order avoids assuming the very infinite hierarchy the variational principle is meant to justify.

🏷️ Key steps and proof outlines

The upper bound is a convexity argument after expanding hierarchical insertions. The lower bound constructs trial messages from finite systems and proves that their joint spin patterns can be separated. “The interpolation upper bound,” “Independent hierarchical messages and the cavity comparison,” and “Normalization, truncation, and completion” make these steps explicit (OpenAI, 2026).

Even arity gives the sign of the interpolation remainder

Interpolate between real interactions of mean $t\alpha N$ and independent cavity interactions of mean $(1-t)\alpha p$ per site. Poisson differentiation replaces a derivative of the count by an insertion of one extra interaction. Subtract the corresponding edge correction. For a replica tree with $k$ leaves, let $P$ be the spatial average of the product of the $k$ interaction factors at a reservoir site, also averaged over one fresh function $f$ shared across those replica factors. Let $B$ be its trial-message analogue, averaged over both that fresh $f$ and the entire independent message tree, including its root. Thus $B$ is deterministic for a fixed replica-tree shape, and both $P$ and $B$ are independent of the inserted interaction coefficient $b$.

The order-$k$ term of the insertion expansion has sign determined by

$$-\frac{{\lambda }^k}{k}\mathbb{E}[(-b{)}^k]\mathbb{E}[P^p-pP{B}^{p-1}+(p-1)B^p],0<\lambda <1.$$

Because $p$ is even, $x\mapsto x^p$ is convex on all of $\mathbb{R}$, so its tangent inequality at $B$ makes the bracket nonnegative. The assumed sign of $\mathbb{E}[(-b{)}^k]$ makes each displayed term nonpositive. This is the precise use of the interaction-class assumptions.

The regularizer $\lambda$ makes the logarithmic expansion absolutely summable. Before expanding, subtract the integrable constant $\theta (1,\dots ,1)$ from every insertion. These constants cancel with coefficients $1,-p,p-1$. The normalized ratios stay between $e^{-2\max |\theta |}$ and $e^{2\max |\theta |}$, permitting dominated convergence as $\lambda \uparrow 1$. No separate integrability assumption on $\log a$ is required.

The site-minus-bond correction comes from Poisson thinning

To compare sizes $N$ and $N+1$, retain interactions using only the old vertices. Their mean is

$${\rho }_N=\alpha (N+1){\left(\frac{N}{N+1}\right)}^p.$$

Interactions using the new vertex exactly once have mean $\alpha p+O(N^{-1})$, while those using it at least twice have mean $O(N^{-1})$. To recover the old $N$-spin model from the common reservoir, add interactions of mean

$$\alpha N-{\rho }_N=\alpha (p-1)+O(N^{-1}).$$

Thus the pressure increment is a new-site insertion minus an old-interaction correction, with the factor $\log 2$ from summing the new spin. This explains the functional’s coefficient $\alpha (p-1)$ directly.

The reservoir also receives $N^{3/4}$ bounded Poisson perturbations in expectation, each factor between $1/2$ and $3/2$. Their total pressure cost per spin is $O(N^{-1/4})$. Coupling the perturbations at neighboring sizes leaves only $O(N^{-1/4})$ unmatched factors in expectation, so the same cavity increment survives. Concentration of their insertion scores is then used in the source’s finite-tree decorrelation argument.

All multioverlaps determine the finite spin pattern

For replicas ${\sigma }^1,\dots ,{\sigma }^k$, define

$$R_S=\frac{1}{N}\sum _i\prod _{a\in S}{\sigma }_i^a,S\subseteq \{1,\dots ,k\},R_{\varnothing }=1.$$

The empirical distribution of their joint pattern at a uniformly chosen site is exactly

$$\pi (e)=2^{-k}\sum _{S\subseteq [k]}{R}_S\prod _{a\in S}{e}_a,e\in \{-1,1{\}}^k.$$

This finite Fourier expansion explains why pair overlaps alone are insufficient. If $\bar{\pi }=\mathbb{E}\pi$, then

$$\Vert {\pi }^{\otimes a}-{\bar{\pi }}^{\otimes a}{\Vert }_{\mathrm{TV}}\le \frac{a}{2}\sum _{S\ne \varnothing }|R_S-\mathbb{E}{R}_S|.$$

The first inequality comes from telescoping product measures; the second from the displayed expansion. Therefore concentration of every multioverlap for the relevant branching trees controls every bounded insertion observable at finitely many sampled sites.

The source obtains the required concentration averaged over branching depths, using bounded perturbations and an induction that delays branches by one level. This averaged result suffices because the logarithmic insertion expansion averages over the same finite tree laws. Irregular or nearly colliding branching levels have total probability $O_k(\eta +L^{-1})$; regular levels are bounded by the averaged variances using Cauchy–Schwarz.

Finite systems encode admissible trial hierarchies

At a leaf of the reservoir hierarchy, set $x=\mathrm{arctanh}⟨{\sigma }_i⟩$ for a uniform site $i$. This is finite because every finite-system spin configuration has positive weight. Take conditional laws backward through the auxiliary levels, producing a random measure of messages, then a measure of those measures, and so on. Conditional independence of branches verifies recursively that this encoding preserves every prescribed one-site replica-tree law. Independent copies of the entire encoding give independent trial labels.

First truncate the number of cavity insertions. For that fixed cutoff, normalize their positive factors as $z(1+D)$ with $|D|\le c<1$, so the logarithmic expansion has a uniform geometric tail. Compare its finitely many terms using the multioverlap estimate, taking system size to infinity at fixed hierarchy depth and then increasing the depth. Remove the expansion and Poisson cutoffs afterward. This proves the matching lower bound without assuming an infinite hierarchy exists.

First moments suffice for removing truncations

If coupled interactions differ by expected sup norm $D$ and fields by expected absolute difference $d$, direct comparison of log weights gives pressure error at most $\alpha D+d$. The trial functional error is at most $\alpha (2p-1)D+d$, uniformly over all depths and trial laws. This uniformity permits taking infima.

Subtracting $\theta (1,\dots ,1)$ first removes the possibly asymmetric multiplier $a$, making the normalized interaction law permutation invariant. Entrywise truncation preserves that symmetry, although it may destroy the factorization. This is harmless because the bounded lower-bound argument uses symmetry but not the factorization. First-moment tails send both errors to zero; undoing the constant shift completes the equality.

See Also

• Family 217 uses hierarchies for a dense Gaussian model but asks about fluctuations below the extensive pressure scale.
• Family 222 explains an overlap variational principle for nonlinear pattern interactions.

🏷️ Manuscripts and verification scope

• The Mézard–Parisi formula for diluted spin glasses — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes the finite-hierarchy variational equality in the stated Panchenko–Talagrand interaction class, under first-moment assumptions. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The Mézard–Parisi formula for diluted spin glasses.