# Spectral gap critical exponent for Glauber dynamics of hierarchical spin models

SRQW01 - Renormalisation in quantum field theory and in stochastic partial differential equations: a gentle introduction and some recent developments

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems at and near a critical point. In our approach, we derive a spectral gap inequality, or more generally a Brascamp—Lieb inequality, for the measure recursively in terms of spectral gap or Brascamp—Lieb 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 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. Co-author: Thierry Bodineau

This talk is part of the Isaac Newton Institute Seminar Series series.