Polynomial mixing of the critical Ising model on sparse Erdos-Renyi graphs

TL;DR

This work proves that at the critical temperature $\beta_c=\tanh^{-1}(d^{-1})$, Glauber dynamics for the Ising model on sparse Erdős–Rényi graphs $G(n,d/n)$ mixes in a polynomial time $t_{\mathrm{mix}}\le n^{D}$ with high probability, for some $D=D(d)$. The authors integrate Chen–Eldan stochastic localization, which yields bulk spectral-gap bounds, with classical results on tree relaxation to control regions with atypically large neighborhoods via a partition into a well-behaved bulk $H$ and a restricted interface $B$. They construct a chain of Markov processes linking the full dynamics to a Restricted Dynamics on $B$, showing polynomial relaxation transfers across the chain, aided by Weitz’s SAW tree and non-backtracking walk analysis. The structural results for $G(n,d/n)$ ensure that, with high probability, the partition properties hold, enabling the layered gap estimates and leading to the main polynomial mixing bound at criticality. This advances understanding of critical Ising behavior on random sparse graphs and provides algorithmic implications for sampling at criticality in irregular graph geometries.

Abstract

We consider the stochastic Ising model on sparse Erdos-Renyi graphs $G(n,d/n)$ with $d>1$ at the critical temperature $β_c=\tanh^{-1}(d^{-1})$ and prove that with high probability, the mixing time is at most polynomial in $n$. Our approach combines the recent stochastic localization framework of Chen and Eldan, which yields spectral gap bounds in the well-behaved bulk of the graph, together with classical results on the relaxation time of Glauber dynamics on trees to handle regions where we cannot apply the Chen-Eldan method directly because of atypically large local neighborhoods.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Definition 1.4
• Remark 1.5
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5