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Superconcentration and chaos in Bernoulli percolation

Van Quyet Nguyen

TL;DR

The paper proves that sublinear fluctuations (superconcentration) of the chemical distance in supercritical Bernoulli percolation are equivalent to chaotic sensitivity of geodesics under small random resampling, extending Chatterjee's framework beyond Gaussian settings. The authors develop a dynamical, edge-centric variance formula using co-influences and introduce a dynamical effective radius to quantify how resampling an edge alters optimal paths; lattice-animal techniques are employed to control the total cost of resampling across edges. By carefully comparing the original and truncated models, they show that the dynamical variance is governed by the overlap of geodesics, yielding a sharp stable-chaotic phase transition characterized by a time scale $\hat{t}_n=\mathrm{Var}({\tilde{\mathrm{D}}}(0,nx))/n$. In particular, they establish that ${\rm Var}({\tilde{\mathrm{D}}}(0,nx))=o(n)$ if and only if there exists a sequence $t_n\to 0$ with ${\mathbb E}[|{\tilde{\pi}}(0,nx)\cap{\tilde{\pi}}^{t_n}(0,nx)|]=o(n)$, and provide sharp bounds linking correlation decay to geodesic overlap, thereby linking fluctuations to chaos in a non-Gaussian percolation context.

Abstract

We study the chemical distance of supercritical Bernoulli percolation on $\mathbb{Z}^d$. Recently, Dembin [Dem22] showed that the chemical distance exhibits sublinear variance, a phenomenon now referred to as superconcentration. In this article, we establish an equivalence between this phenomenon and chaotic behavior of geodesics under small perturbations of the configuration, thereby confirming Chatterjee's general principle relating anomalous fluctuations to chaos in the context of Bernoulli percolation. Our methods rely on a dynamical version of the effective radius, refining the notion first proposed in [CN25], in order to measure the co-influence of a given edge whose weight may be infinite. Together with techniques from the theory of lattice animals, this approach allows us to quantify the total co-influence of edges in terms of the overlap between original and perturbed geodesics.

Superconcentration and chaos in Bernoulli percolation

TL;DR

The paper proves that sublinear fluctuations (superconcentration) of the chemical distance in supercritical Bernoulli percolation are equivalent to chaotic sensitivity of geodesics under small random resampling, extending Chatterjee's framework beyond Gaussian settings. The authors develop a dynamical, edge-centric variance formula using co-influences and introduce a dynamical effective radius to quantify how resampling an edge alters optimal paths; lattice-animal techniques are employed to control the total cost of resampling across edges. By carefully comparing the original and truncated models, they show that the dynamical variance is governed by the overlap of geodesics, yielding a sharp stable-chaotic phase transition characterized by a time scale . In particular, they establish that if and only if there exists a sequence with , and provide sharp bounds linking correlation decay to geodesic overlap, thereby linking fluctuations to chaos in a non-Gaussian percolation context.

Abstract

We study the chemical distance of supercritical Bernoulli percolation on . Recently, Dembin [Dem22] showed that the chemical distance exhibits sublinear variance, a phenomenon now referred to as superconcentration. In this article, we establish an equivalence between this phenomenon and chaotic behavior of geodesics under small perturbations of the configuration, thereby confirming Chatterjee's general principle relating anomalous fluctuations to chaos in the context of Bernoulli percolation. Our methods rely on a dynamical version of the effective radius, refining the notion first proposed in [CN25], in order to measure the co-influence of a given edge whose weight may be infinite. Together with techniques from the theory of lattice animals, this approach allows us to quantify the total co-influence of edges in terms of the overlap between original and perturbed geodesics.
Paper Structure (16 sections, 24 theorems, 196 equations, 1 figure)

This paper contains 16 sections, 24 theorems, 196 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and main result
  3. Sketch of the proofs
  4. Proof of Theorem \ref{['Theorem: variance and chaos']} and \ref{['theorem: main']} via Theorem \ref{['Theorem: a dynamic formula for Dkm']}
  5. Structure of the paper and some notations
  6. Ingredients for the proof of main results
  7. Preliminary on supercritical percolation
  8. The co-influence bound via effective dynamic radius
  9. Estimates for lattice animals
  10. Lower bound for the intersection of geodesics
  11. Dynamical variance inequalities for first truncated passage time
  12. A dynamical formula for variance
  13. Proof of Theorem \ref{['Theorem: a dynamic formula for Dkm']}
  14. The comparison of ${\tilde{\mathrm{D}}}^t(0,nx)$ and ${\tilde{\mathrm{T}}}_M^t(0,nx)$
  15. Dynamical formula for variance: Proof of Lemma \ref{['Lemma: covariacne formula']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists a positive constant $C$ such that for all $n \geq 1$ and $x \in {\mathbb Z}^d$, and for all $\alpha> 0$, Consequently, as $n \to \infty$, we obtain

Figures (1)

  • Figure 1: Illustration of the event ${\mathcal{E}}_6^t$

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['Theorem: variance and chaos']}
  • proof : Proof of Theorem \ref{['theorem: main']}
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 26 more