A universal cutoff phenomenon for mean-field exchange models

TL;DR

This paper proves a universal cutoff phenomenon in high-dimensional mean-field exchange models, showing sharp convergence to equilibrium with a Gaussian profile, characterized by the size-biased redistribution variable.

Contribution

It introduces a unified framework for analyzing convergence in diverse mean-field exchange models and establishes a universal cutoff phenomenon with explicit mixing time and profile.

Findings

Universal cutoff phenomenon at a specific mixing time

Gaussian limiting profile for convergence

Characterization of mixing time via size-biased redistribution variable

Abstract

We study a broad class of high-dimensional mean-field exchange models, encompassing both noisy and singular dynamics, along with their dual processes. This includes a generalized version of the averaging process as well as some non-reversible extensions of classical exchange dynamics, such as the flat Kac model. Within a unified framework, we analyze convergence to stationarity from worst-case initial data in Wasserstein distance. Our main result establishes a universal cutoff phenomenon at an explicit mixing time, with a precise window and limiting Gaussian profile. The mixing time and profile are characterized in terms of the logarithm of the size-biased redistribution random variable, thus admitting a natural entropic interpretation.