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A phase transition for repeated averages
Sourav Chatterjee, Persi Diaconis, Allan Sly, Lingfu Zhang · 2019-11-07 · via math.PR updates on arXiv.org

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.

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