Trickle-Down in Localization Schemes and Applications

TL;DR

This paper introduces a generalized trickle-down framework for high-dimensional expanders, leading to improved results in Markov chain mixing times and sampling algorithms across various models.

Contribution

It formulates a generalized trickle-down equation within linear-tilt localization schemes, enhancing mixing time bounds and sampling algorithms for complex probabilistic models.

Findings

Improved rapid mixing thresholds for Glauber dynamics in spin glasses

Enhanced mixing results for Langevin dynamics in the O(N) model

Near-linear time sampling algorithms for Ising models on expanders

Abstract

Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems -- for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include improved mixing results for the Langevin dynamics in the $O(N)$ model, and near-linear time sampling algorithms for the antiferromagnetic and fixed-magnetization Ising models on expanders. For the latter application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials.