Spectral gap critical exponent for Glauber dynamics of hierarchical spin models

TL;DR

This paper introduces a renormalisation group method to analyze the spectral gap decay of Glauber dynamics in hierarchical spin models at criticality, revealing polynomial decay with specific corrections.

Contribution

It develops a recursive spectral gap inequality approach and applies it to several hierarchical models at critical points, providing new insights into their dynamics.

Findings

Spectral gap decays polynomially with system size.

Logarithmic correction for the $| ext{varphi}|^4$ model.

Scaling limit matches free field dynamics.

Abstract

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the $4$-dimensional $n$-component $|\varphi|^4$ model at the critical point and its approach from the high temperature side, and of the $2$-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz--Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the $|\varphi|^4$ model), the scaling limit of these models in equilibrium.