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Communications in Mathematical Physics

, Volume 335, Issue 3, pp 1287–1322 | Cite as

Cutoff for the East Process

  • S. Ganguly
  • E. LubetzkyEmail author
  • F. Martinelli
Article

Abstract

The East process is a 1d kinetically constrained interacting particle system, introduced in the physics literature in the early 1990s to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on L sites has order L. We complement that result and show cutoff with an \({O(\sqrt{L})}\)-window.

The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an \({O(\sqrt{L})}\)-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure ν, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to ν, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen (‘82) implies a CLT for the location of the front, yielding the cutoff result.

Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an O(1)-window.

Keywords

Invariant Measure Graphical Construction Maximal Coupling Coin Toss Reversible Markov Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Courant InstituteNew York UniversityNew YorkUSA
  3. 3.Dip. Matematica & FisicaUniversità Roma TreRomaItaly

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