January 2022 A phase transition for repeated averages
Sourav Chatterjee, Persi Diaconis, Allan Sly, Lingfu Zhang
Author Affiliations +
Ann. Probab. 50(1): 1-17 (January 2022). DOI: 10.1214/21-AOP1526


Let x1,,xn be a fixed sequence of real numbers. At each stage, pick two indices I and J uniformly at random, and replace xI, xJ by (xI+xJ)/2, (xI+xJ)/2. Clearly, all the coordinates converge to (x1++xn)/n. We determine the rate of convergence, establishing a sharp “cutoff” transition answering a question of Jean Bourgain.

Funding Statement

Sourav Chatterjee’s research was partially supported by NSF Grant DMS-1855484. Persi Diaconis’s research was partially supported by NSF Grant DMS-0804324. Allan Sly’s research was partially supported by NSF Grant DMS-1855527, Simons Investigator Grant and a MacArthur Fellowship.


We thank David Aldous, Laurent Miclo, Evita Nestoridi and Perla Sousi for comments. Our revival of this project stems from a quantum computing question of Ramis Movassagh—roughly, to develop a convergence result with [ 1212 1212] replaced by the parallel quantum spin operator—we hope to develop this in joint work with him. Finally, we acknowledge the great, late Jean Bourgain. He sent us a typed draft of a preliminary manuscript. Alas, 30 years later this proved difficult to find. We will be grateful if a copy can be located.


Download Citation

Sourav Chatterjee. Persi Diaconis. Allan Sly. Lingfu Zhang. "A phase transition for repeated averages." Ann. Probab. 50 (1) 1 - 17, January 2022. https://doi.org/10.1214/21-AOP1526


Received: 1 August 2020; Revised: 1 March 2021; Published: January 2022
First available in Project Euclid: 23 February 2022

MathSciNet: MR4385355
zbMATH: 1485.60069
Digital Object Identifier: 10.1214/21-AOP1526

Primary: 60J05 , 60J20

Keywords: convergence rate , Cutoff phenomenon , Markov chain

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 1 • January 2022
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