On sampling diluted Spin Glasses using Glauber dynamics
Charilaos Efthymiou, Kostas Zampetakis
TL;DR
This work analyzes sampling from Edwards-Anderson spin-glass distributions on sparse Erdős–Rényi graphs via Glauber dynamics. The authors introduce aggregate influence and a path-weighted framework, then construct a block-partition of $G(n,d/n)$ to hide heavy vertices inside small blocks and prove rapid mixing for the block dynamics using path coupling. By a comparison argument, they transfer this rapid mixing to the standard Glauber dynamics, obtaining a mixing time of $T_{Mix}=O\left(n^{2+\frac{3}{\log^2 d}}\right)$ for inverse temperatures $\beta$ with $\mathbb{E}[\mathcal{A}(w)]<1/d$ (and hence finite correlation decay). Their analysis avoids spectral-independence techniques, instead leveraging classical path coupling adapted to aggregate-influence weights, block-structures, and structural properties of $G(n,d/n)$. The results extend to sub-Gaussian couplings and highlight a near-optimal regime where dilution and disorder permit efficient sampling, with a conjectured lower-bound behavior suggesting $T_{Mix}$ could be as large as $n\exp(\Theta(\sqrt{\log n}))$ in the heavy-tail regime. Overall, the paper advances our understanding of MCMC performance for diluted spin-glass models, bridging geometry and disorder through a novel aggregation-based contraction framework.
Abstract
Spin-glasses are Gibbs distributions that have been studied in CS for many decades. Recently, they have gained renewed attention as they emerge naturally in learning, inference, optimisation etc. We consider the Edwards-Anderson (EA) spin-glass distribution at inverse temperature $β$ when the underlying graph is an instance of $G(n,d/n)$. This is the random graph on $n$ vertices where each edge appears independently with probability $d/n$ and $d=Θ(1)$. We study the problem of approximate sampling from this distribution using Glauber dynamics. For a range of $β$ that depends on $d$ and for typical instances of the EA model on $G(n,d/n)$, we show that the corresponding Glauber dynamics exhibits mixing time $O(n^{2+\frac{3}{\log^2 d}})$. The range of $β$ for which we obtain our rapid-mixing results correspond to the expected influence being $<1/d$; we conjecture that this is the best possible. Unlike the mean-field spin-glasses, where the problem has been studied before, the diluted case has not. We utilise the well-known path-coupling technique. In the standard Glauber dynamics on $G(n,d/n)$, one has to deal with the so-called effect of high degree vertices. Here, rather than considering degrees, it is more natural to use a different measure on the vertices called aggregate influence. We build on the block-construction approach proposed by [Dyer et al. 2006] to circumvent the problem of high-degree vertices. Specifically, we first establish rapid mixing for an appropriately defined block-dynamics. We design this dynamics such that vertices of large aggregate influence are placed deep inside their blocks. Then, we obtain rapid mixing for the Glauber dynamics utilising a comparison argument.