Small noise spectral gap asymptotics for a large system of nonlinear diffusions

TL;DR

This paper analyzes the spectral gap behavior of large, strongly coupled nonlinear diffusions with double-well potentials, providing bounds and conditions for asymptotic optimality in the small temperature limit.

Contribution

It offers new bounds for the spectral gap of large coupled diffusions and connects these results to semiclassical analysis of the Witten Laplacian.

Findings

Upper bound given by an Eyring-Kramers-type formula

Lower bound proven for the logarithmic Sobolev constant

Conditions established for asymptotic optimality of the bounds

Abstract

We study the $L^2$ spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen-Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size. The upper bound is given by an Eyring-Kramers-type formula. The lower bound is proven to hold also for the logarithmic Sobolev constant. We establish a sufficient condition for the asymptotic optimality of the upper bound and show that this condition is fulfilled under suitable assumptions on the growth of the system size. Our results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.