Cutoff for congestion dynamics and related generalized exclusion processes

TL;DR

This paper analyzes the cutoff phenomenon in congestion dynamics and related Markov chains, revealing precise mixing times and contrasting behaviors in different sampling scenarios.

Contribution

It establishes cutoff times for Glauber dynamics in congestion models and highlights cases where cutoff does not occur in M-convex set sampling.

Findings

Glauber dynamics exhibits cutoff at (1/2)n log n time

Unlabeled version exhibits cutoff at (1/2)(1-ρ)n log n

Some M-convex set sampling chains do not show cutoff

Abstract

We consider congestion dynamics with $n$ players and $Q$ resources under the constraint that the number of each resource is $\kappa$ and that $n<\kappa Q$ in the regime that $n$ and $\kappa$ diverge but $Q$ is fixed with $n=\lfloor{\rho \kappa Q\rfloor}$ for a fixed constant $\rho \in (0, 1/2]$. We show that the Glauber dynamics and its unlabeled version exhibit cutoff at time $(1/2)n \log n$ and $(1/2)(1-\rho)n\log n$ in total variation respectively. The unlabeled version is a special case of natural Markov chains for sampling from log M-concave distributions. We also show that a family of Markov chains for uniform sampling on M-convex sets does not necessarily exhibit cutoff.