Convergence to equilibrium for density dependent Markov jump processes

TL;DR

This paper studies how density dependent Markov chains rapidly approach equilibrium, revealing a cutoff phenomenon with a sharp transition and a constant-size window, especially as the system size grows large.

Contribution

It proves the existence of a cutoff phenomenon with an optimal constant window for density dependent Markov jump processes approaching equilibrium.

Findings

Demonstrates the cutoff phenomenon in Markov chains with density dependence.

Identifies the cutoff window as asymptotically constant in size.

Links the mixing time to the solution of the drift differential equations.

Abstract

We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb Z}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb X}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb X}^N$, started in some given state, approaches its equilibrium distribution~$\pi^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt N$ from their fixed point. However, the length of the time interval over which the total variation distance between ${\mathcal L} ({\mathbb X}^N(t))$ and its equilibrium distribution~$\pi^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size.