A phase transition for repeated averages

Let $x_1,\dots ,x_n$ 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 $(x_I+x_J)\mathrm{/}2$, $(x_I+x_J)\mathrm{/}2$. Clearly all the coordinates converge to $(x_1+\cdots +x_n)\mathrm{/}n$. We determine the rate of convergence, establishing a sharp “cutoff” transition, answering a question of Jean Bourgain.