A phase transition for repeated averages

Abstract

Let ${\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{n}}$ be a fixed sequence of real numbers. At each stage, pick two indices I and J uniformly at random, and replace ${\mathit{x}}_{\mathit{I}}$, ${\mathit{x}}_{\mathit{J}}$ by $({\mathit{x}}_{\mathit{I}}+{\mathit{x}}_{\mathit{J}})/2$, $({\mathit{x}}_{\mathit{I}}+{\mathit{x}}_{\mathit{J}})/2$. Clearly, all the coordinates converge to $({\mathit{x}}_1+\cdots +{\mathit{x}}_{\mathit{n}})/\mathit{n}$. We determine the rate of convergence, establishing a sharp “cutoff” transition answering a question of Jean Bourgain.