On sampling diluted Spin-Glasses with unbounded interactions

2026-03-23

AI Summary

This work investigates efficient sampling for the two-spin glass model with Gaussian unbounded couplings on sparse random graphs \( G(n, d/n) \). Addressing the challenges posed by unbounded degrees and interactions, it introduces— for the first time—the stochastic localization method to sparse spin glass systems, combining it with Glauber dynamics and probabilistic graphical model techniques to analyze the mixing time of the associated Markov chain. The study establishes that, for inverse temperature \( \beta < 1/(4\sqrt{d}) \), the mixing time on typical instances is \( O(n^{1+25/\sqrt{\log d}}) \), significantly improving upon the previously known rapid mixing threshold for the Viana–Bray model.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Spin-glasses are natural Gibbs distributions that have been studied in Theoretical CS for many decades. Recently, they have been gaining attention from the community as they emerge naturally in neural computation and learning, network inference, optimisation and other areas. We study the problem of efficiently sampling from spin-glass distributions when the underlying graph is a typical instance of $G(n,d/n)$, i.e., the random graph on $n$ vertices such that each edge appears independently with probability $d/n$, and $d=Θ(1)$. Our focus is on the 2-spin model at inverse temperature $β$. We consider this distribution to be one of the most interesting case of spin-glasses, and one of the most challenging to analyse, since its Gaussian couplings give rise to unbounded interaction. We employ the well-known Glauber dynamics to sample from the aforementioned distribution. We show that for the typical instances of the 2-spin model on $G(n,d/n)$, the mixing time of Glauber dynamics is $O\left(n^{1+\frac{25}{\sqrt{\log d}}}\right)$, for any $β<\frac{1}{4\sqrt{d}}$. Our results can also be adapted for the case of spin-glass distributions with bounded interactions. In that respect, we obtain rapid mixing of Glauber dynamics for the Viana-Bray model on $G(n,d/n)$ when $β<\frac{1}{4\sqrt{d}}$. This improves on the current best bound which is $β<\frac{0.18}{\sqrt{d}}$. We utilise stochastic localisation, and in particular, we build and improve on the scheme introduced in [Liu, Mohanty, Rajaraman and Wu: FOCS 2024]. This is the first time that stochastic localisation is used for diluted spin-glasses, where both degrees and interactions can be unbounded.

Problem

• spin-glasses
• sampling
• unbounded interactions
• diluted graphs
• Glauber dynamics

Innovation

• stochastic localisation
• diluted spin-glasses
• unbounded interactions
• Glauber dynamics
• mixing time