Metropolis algorithm and equienergy sampling for two mean field spin systems

TL;DR

This paper investigates how modifications to the proposal chain in the Metropolis algorithm can significantly improve mixing times for mean-field spin systems, reducing spectral gap decay from exponential to polynomial in system size.

Contribution

It introduces a variant of the proposal chain that maintains computational cost while improving spectral gap decay from exponential to polynomial for certain mean-field models.

Findings

Modified proposal chain improves spectral gap decay rate

Method maintains similar computational cost as standard Metropolis

Applicable to mean-field Ising and Blume-Emery-Griffiths models

Abstract

In this paper we study the Metropolis algorithm in connection with two mean--field spin systems, the so called mean--field Ising model and the Blume--Emery--Griffiths model. In both this examples the naive choice of proposal chain gives rise, for some parameters, to a slowly mixing Metropolis chain, that is a chain whose spectral gap decreases exponentially fast (in the dimension $N$ of the problem). Here we show how a slight variant in the proposal chain can avoid this problem, keeping the mean computational cost similar to the cost of the usual Metropolis. More precisely we prove that, with a suitable variant in the proposal, the Metropolis chain has a spectral gap which decreases polynomially in 1/N. Using some symmetry structure of the energy, the method rests on allowing appropriate jumps within the energy level of the starting state.