Fast Mixing in Sparse Random Ising Models

TL;DR

This work analyzes fast sampling from Ising models with sparse random interactions, notably on Erdős–Rényi graphs and stochastic block models. The authors develop a two-pronged approach: a graph decomposition into a tame bulk and a controllable near-forest, and a measure-decomposition via a Hubbard–Stratonovich transform together with stochastic localization to reduce to simpler, tractable Ising models. They prove modified log-Sobolev inequalities (MLSIs) and entropy-conservation along the localization path, yielding nearly-linear mixing times $t_{ ext{mix}}=n^{1+o(1)}$ for diluted spin glasses in the $G(n,d/n)$ regime with $β \\leq C/\\sqrt{d}$, independent of $n$, and extend these results to centered SBMs. A companion paper leverages the centered-adjacency results to establish community-recovery performance for the SBM via Glauber dynamics, highlighting the method’s relevance to inference problems. Overall, the paper provides a robust framework for fast sampling in sparse random Ising models and sharpens the boundary between computational tractability and hardness in average-case regimes.

Abstract

Motivated by the community detection problem in Bayesian inference, as well as the recent explosion of interest in spin glasses from statistical physics, we study the classical Glauber dynamics for sampling from Ising models with sparse random interactions. It is now well-known that when the interaction matrix has spectral diameter less than $1$, Glauber dynamics mixes in $O(n\log n)$ steps. Unfortunately, such criteria fail dramatically for interactions supported on arguably the most well-studied sparse random graph: the Erdős--Rényi random graph $G(n,d/n)$, due to the presence of almost linearly many outlier eigenvalues of unbounded magnitude. We prove that for the \emph{Viana--Bray spin glass}, where the interactions are supported on $G(n,d/n)$ and randomly assigned $\pmβ$, Glauber dynamics mixes in $n^{1+o(1)}$ time with high probability as long as $β\le O(1/\sqrt{d})$, independent of $n$. We further extend our results to random graphs drawn according to the $2$-community stochastic block model, as well as when the interactions are given by a "centered" version of the adjacency matrix. The latter setting is particularly relevant for the inference problem in community detection. Indeed, we use this to show that Glauber dynamics succeeds at recovering communities in the stochastic block model in a companion paper [LMR+24]. The primary technical ingredient in our proof is showing that with high probability, a sparse random graph can be decomposed into two parts -- a \emph{bulk} which behaves like a graph with bounded maximum degree and a well-behaved spectrum, and a \emph{near-forest} with favorable pseudorandom properties. We then use this decomposition to design a localization procedure that interpolates to simpler Ising models supported only on the near-forest, and then execute a pathwise analysis to establish a modified log-Sobolev inequality.

Figures (4)

• Figure 1: An illustration of our graph decomposition (\ref{['lem:graph-decomp-overview']}) for a sparse SBM with average degree $d = 3$ and $\varepsilon = 0.5$. The bulk $B$ (green) consists solely of vertices with degree at most $d(1+\varepsilon) = 4.5$. The near-forest $H$ (shaded blue circles) comprise the local neighborhoods of high-degree vertices (enlarged for emphasis) in $\boldsymbol{G}$. The local neighborhoods of nearby high-degree vertices can merge, as shown by the two blue circles in the top right. Nevertheless, with high probability, all the connected components in $H$ are trees with at most one additional edge.
• Figure 2: A flow chart outlining our proof of an MLSI for diluted spin glasses.
• Figure 3: A flow chart outlining our proof of an MLSI for the centered adjacency matrix. The main difference in the structure of the proof compared to \ref{['fig:spinglass-flowchart']} are the three boxes on the bottom right, which handle the interactions present on nonedges of the graph due to centering.
• Figure 4: A depiction of the sets $T_1, T_2, T', S$, and $B_{\ell}(u)$. The event of interest in \ref{['lem:gain-if-tree-excess']} is when the random set of interest $B_{\ell}(u)$ (shaded blue) intersects the two target sets $T_1$ (solid purple) and $T_2$ (solid green). The other two sets of vertices $S$ and $T'$ cover the additional edges that need to be conditioned on to apply the lemma inductively to bound the length of the path $u_1\ldots u_k$ in the cluster graph. In our setting, all the sets involved are unions of balls centered at the vertices $u_i$.

Theorems & Definitions (134)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 2.1: Informal version of \ref{['thm:main-mixing']}
• Remark 2.2
• Lemma 2.2
• Definition 2.3: $(\Delta, D)$-pseudorandomness, informal
• Theorem 2.4: Informal version of \ref{['thm:mlsi-bdd-growth-tree']}
• Lemma 2.4
• Definition 3.1: Glauber dynamics