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

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 Schrödinger operator.

44 pages, 10 figures. This preprint includes the previous results for the mean-field $\mathcal{O}(n)$-model and contains new results (sharp spectral gap for the critical temperature-case) with improved presentation. The second part of the previous version, on the chain of oscillators, will be outsourced into an independent preprint