The spectrum and convergence rates of exclusion and interchange processes on the complete graph

TL;DR

This paper provides an elementary method to determine the spectrum and convergence rates of exclusion and interchange processes on the complete graph, simplifying and strengthening existing results on their mixing times.

Contribution

It introduces a simple, elementary approach to exactly find the spectrum of the exclusion process and bounds for the interchange process on the complete graph.

Findings

Exact eigenvalues and multiplicities for the exclusion process

Closed-form expression for L^2 distance from stationarity

Upper and lower bounds on convergence rates

Abstract

We give a short and completely elementary method to find the full spectrum of the exclusion process and a nicely limited superset of the spectrum of the interchange process (a.k.a.\ random transpositions) on the complete graph. In the case of the exclusion process, this gives a simple closed form expression for all the eigenvalues and their multiplicities. This result is then used to give an exact expression for the distance in $ L^2 $ from stationarity at any time and upper and lower bounds on the convergence rate for the exclusion process. In the case of the interchange process, upper and lower bounds are similarly found. Our results strengthen or reprove all known results of the mixing time for the two processes in a very simple way.