A phase transition for repeated averages

TL;DR

This paper investigates the convergence rate of a stochastic averaging process on fixed real sequences, revealing a sharp phase transition in the convergence behavior, thus answering a question posed by Jean Bourgain.

Contribution

It precisely characterizes the convergence rate and identifies a sharp cutoff transition in the repeated averaging process.

Findings

Identifies a phase transition in the convergence rate.

Provides a sharp cutoff phenomenon in the averaging process.

Answers a longstanding question of Jean Bourgain.

Abstract

Let $x_1,\ldots,x_n$ be a fixed sequence of real numbers. At each stage, pick two indices $I$ and $J$ uniformly at random and replace $x_I$, $x_J$ by $(x_I+x_J)/2$, $(x_I+x_J)/2$. Clearly all the coordinates converge to $(x_1+\cdots+x_n)/n$. We determine the rate of convergence, establishing a sharp "cutoff" transition, answering a question of Jean Bourgain.