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Critical speeding-up in dynamical percolation

Eren Metin Elci, Timothy M. Garoni

Abstract

We study the autocorrelation time of the size of the cluster at the origin in discrete-time dynamical percolation. We focus on binary trees and high-dimensional tori, and show in both cases that this autocorrelation time is linear in the volume in the subcritical regime, but strictly sublinear in the volume at criticality. This establishes rigorously that the cluster size at the origin in these models exhibits critical speeding-up. The proofs involve controlling relevant Fourier coefficients. In the case of binary trees, these Fourier coefficients are studied explicitly, while for high-dimensional tori we employ a randomised algorithm argument introduced by Schramm and Steif in the context of noise sensitivity.

Critical speeding-up in dynamical percolation

Abstract

We study the autocorrelation time of the size of the cluster at the origin in discrete-time dynamical percolation. We focus on binary trees and high-dimensional tori, and show in both cases that this autocorrelation time is linear in the volume in the subcritical regime, but strictly sublinear in the volume at criticality. This establishes rigorously that the cluster size at the origin in these models exhibits critical speeding-up. The proofs involve controlling relevant Fourier coefficients. In the case of binary trees, these Fourier coefficients are studied explicitly, while for high-dimensional tori we employ a randomised algorithm argument introduced by Schramm and Steif in the context of noise sensitivity.
Paper Structure (12 sections, 9 theorems, 120 equations)

This paper contains 12 sections, 9 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Definitions and results
  5. Dynamical percolation and autocorrelations
  6. Results
  7. Spectral representations
  8. Autocorrelations
  9. Randomised bit-query algorithms
  10. Binary Trees
  11. High-dimensional tori
  12. Proof of the revealment-predictability theorem

Key Result

Lemma 2.1

Fix $p\in(0,1)$. Let $B_n$ be an increasing sequence of finite sets, and let $f_n:S^{B_n}\to\mathbb{R}$ be a sequence of non-constant functions. Then as $n\to\infty$, for any positive integer sequence $s_n$ satisfying $s_n=\omega(1)$ we have

Theorems & Definitions (22)

  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 3.1
  • Lemma 3.2
  • proof : Proof of lemma \ref{['lem:discrete time vs continuous time rho asymptotics']}
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Theorem 3.6
  • ...and 12 more