Sampling the Fermi statistics and other conditional product measures

TL;DR

This paper introduces an efficient Markov chain algorithm for sampling Fermi-Dirac statistics and other constrained product measures, providing explicit bounds on mixing times that are independent of disorder.

Contribution

It develops a general framework for analyzing mixing times of Markov chains sampling constrained product measures, including Fermi statistics and exclusion processes.

Findings

Explicit upper bounds on mixing times of the Markov chain.

The bounds are uniform over temperature, energy levels, and degeneracies.

Extension of results to non-homogeneous log-concave measures.

Abstract

Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km(ln k) for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non-homogeneous) products of ultra log-concave measures (like binomial or Poisson laws) with a global constraint. As a consequence of this general result we also obtain a disorder-independent upper bound on the mixing time of a simple exclusion process on the complete graph with site disorder. This general result is based on an elementary coupling argument and extended to (non-homogeneous) products of log-concave measures.