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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Information percolation and cutoff for the stochastic Ising model
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by Eyal Lubetzky and Allan Sly
J. Amer. Math. Soc. 29 (2016), 729-774
Published electronically: September 29, 2015


We introduce a new framework for analyzing Glauber dynamics for the Ising model. The traditional approach for obtaining sharp mixing results has been to appeal to estimates on spatial properties of the stationary measure from within a multi-scale analysis of the dynamics. Here we propose to study these simultaneously by examining “information percolation” clusters in the space-time slab.

Using this framework, we obtain new results for the Ising model on $(\mathbb Z/n\mathbb Z)^d$ throughout the high temperature regime: total-variation mixing exhibits cutoff with an $O(1)$-window around the time at which the magnetization is the square root of the volume. (Previously, cutoff in the full high temperature regime was only known in dimensions $d\leq 2$, and only with an $O(\log \log n)$-window.)

Furthermore, the new framework opens the door to understanding the effect of the initial state on the mixing time. We demonstrate this on the 1D Ising model, showing that starting from the uniform (“disordered”) initial distribution asymptotically halves the mixing time, whereas almost every deterministic starting state is asymptotically as bad as starting from the (“ordered”) all-plus state.

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Bibliographic Information
  • Eyal Lubetzky
  • Affiliation: Courant Institute, New York University, 251 Mercer Street, New York, New York 10012
  • MR Author ID: 787713
  • ORCID: 0000-0002-2281-3542
  • Email:
  • Allan Sly
  • Affiliation: Department of Statistics, UC Berkeley, Berkeley, California 94720
  • MR Author ID: 820461
  • Email:
  • Received by editor(s): July 14, 2014
  • Received by editor(s) in revised form: May 7, 2015
  • Published electronically: September 29, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 729-774
  • MSC (2010): Primary 60J27, 82C20; Secondary 60K35, 60B10
  • DOI:
  • MathSciNet review: 3486171