Spectral gap in mean-field $\mathcal O(n)$-model

TL;DR

This paper investigates how the spectral gap of the Ginzburg-Landau dynamics generator varies with the number of particles in mean-field f(n)-models, using renormalization and semiclassical techniques.

Contribution

It provides a detailed analysis of the spectral gap dependence on particle number for f(n)-models with mean-field interaction and magnetic field, including at critical temperature.

Findings

Spectral gap behavior characterized for all f(n)-models.

Application of renormalization and semiclassical methods.

Insights into eigenvalue-spacing of related Schrf6dinger operators.

Abstract

We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all \emph{$\mathcal O(n)$-models} with mean-field interaction and magnetic field, below and at the critical temperature on the number $N$ of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schr\"odinger operator.