Repeated Block Averages: entropic time and mixing profiles

TL;DR

This paper analyzes the convergence and cutoff phenomena of randomized averaging processes over the n-simplex, characterizing how block size distribution influences mixing times and providing explicit profiles for the cutoff behavior.

Contribution

It offers sharp conditions for cutoff emergence, characterizes cutoff windows, and details the impact of block size distribution on convergence in averaged processes.

Findings

Cutoff phenomenon occurs under specific block size distributions.

Explicit Gaussian cutoff profile is derived.

Large block sizes prevent cutoff, leading to different convergence behavior.

Abstract

We consider randomized dynamics over the $n$-simplex, where at each step a random set, or block, of coordinates is evenly averaged. When all blocks have size 2, this reduces to the repeated averages studied in [CDSZ22], a version of the averaging process on a graph [AL12]. We study the convergence to equilibrium of this process as a function of the distribution of the block size, and provide sharp conditions for the emergence of the cutoff phenomenon. Moreover, we characterize the size of the cutoff window and provide an explicit Gaussian cutoff profile. To complete the analysis, we study in detail the simplified case where the block size is not random. We show that the absence of a cutoff is equivalent to having blocks of size $n^{\Omega(1)}$, in which case we provide a convergence in distribution for the total variation distance at any given time, showing that, on the proper time scale, it remains constantly 1 up to an exponentially distributed random time, after which it decays following a Poissonian profile.