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Limiting distribution of the chemical distance in high dimensional critical percolation

Shirshendu Chatterjee, Pranav Chinmay, Jack Hanson, Philippe Sosoe

TL;DR

The paper identifies the limiting distribution of the chemical distance in high-dimensional critical Bernoulli percolation conditioned on connectivity, showing that when scaled by a constant times the square of the Euclidean distance, the distance converges to a density equal to the hitting-time distribution of a Brownian motion in $\mathbb{R}^d$ conditioned to hit a unit vector. The approach hinges on a moment method for additive quantities over pivotal edges, the construction of local variables, and a decoupling via convergence to the incipient infinite cluster (IIC). Central to the analysis are precise moment estimates separating far- and near-regimes and a detailed control of bubbles, including large-bubble truncations and truncation in distance across bubbles. The results leverage known two-point function asymptotics and the IIC framework to derive explicit limiting constants and the Brownian-type limit, establishing a robust scaling picture for critical high-dimensional percolation. The findings illuminate the universal nature of the scaling limit and connect percolation geometry to stochastic processes, with potential extensions to joint-distance distributions and scaling limits of cluster measures.

Abstract

We identify the asymptotic distribution of the chemical distance in high-dimensional critical Bernoulli percolation. Namely, we show that the distance between the origin and a distant vertex conditioned to lie in the cluster of the origin converges in distribution when rescaled by a multiple the square of the Euclidean distance. The limiting distribution has an explicit density and coincides with the distribution of the time for a Brownian motion in $\mathbb{R}^d$ conditioned to hit a given unit vector to reach its target. Our result follows from a general moment computation for quantities that have an additive structure across the pivotal edges on a long-range connection in percolation. In addition to the number of pivotal edges in a long connection, this also includes the effective resistance. The existence of the incipient infinite cluster limit, in a form recently established, plays a key role in the derivation of our results.

Limiting distribution of the chemical distance in high dimensional critical percolation

TL;DR

The paper identifies the limiting distribution of the chemical distance in high-dimensional critical Bernoulli percolation conditioned on connectivity, showing that when scaled by a constant times the square of the Euclidean distance, the distance converges to a density equal to the hitting-time distribution of a Brownian motion in conditioned to hit a unit vector. The approach hinges on a moment method for additive quantities over pivotal edges, the construction of local variables, and a decoupling via convergence to the incipient infinite cluster (IIC). Central to the analysis are precise moment estimates separating far- and near-regimes and a detailed control of bubbles, including large-bubble truncations and truncation in distance across bubbles. The results leverage known two-point function asymptotics and the IIC framework to derive explicit limiting constants and the Brownian-type limit, establishing a robust scaling picture for critical high-dimensional percolation. The findings illuminate the universal nature of the scaling limit and connect percolation geometry to stochastic processes, with potential extensions to joint-distance distributions and scaling limits of cluster measures.

Abstract

We identify the asymptotic distribution of the chemical distance in high-dimensional critical Bernoulli percolation. Namely, we show that the distance between the origin and a distant vertex conditioned to lie in the cluster of the origin converges in distribution when rescaled by a multiple the square of the Euclidean distance. The limiting distribution has an explicit density and coincides with the distribution of the time for a Brownian motion in conditioned to hit a given unit vector to reach its target. Our result follows from a general moment computation for quantities that have an additive structure across the pivotal edges on a long-range connection in percolation. In addition to the number of pivotal edges in a long connection, this also includes the effective resistance. The existence of the incipient infinite cluster limit, in a form recently established, plays a key role in the derivation of our results.

Paper Structure

This paper contains 36 sections, 31 theorems, 353 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Result
  3. Motivation
  4. Outline of the proof
  5. Preliminaries
  6. Definitions and basic estimates
  7. Additive Quantities
  8. Bubbles
  9. Additivity
  10. Local variables
  11. Limit Theorem for local variables
  12. Moment estimates
  13. Proof of Theorem \ref{['mainthm']}
  14. Far-regime
  15. Recursive setup for the far-regime
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\mathbf{e}_1:=(1,0,\ldots,0)$ be the unit vector in the first coordinate direction in $\mathbb{Z}^d$. When there is no risk of confusion, we denote the origin $(0,\ldots,0)\in \mathbb{Z}^d$ simply by $0$. Define the following quantities: Let and suppose eqn: two-pt holds. (By HHS and FH, this last assumption is satisfied whenever $d\geq 11$.) Then there are constants $\mathbf{c}$, $\beta$,

Figures (2)

  • Figure 1: An illustration of the cluster $\zeta_{k-1}^*$
  • Figure 2: An illustration of the proof of Lemma \ref{['lem: distances-differ']}

Theorems & Definitions (74)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 2
  • Lemma 1
  • ...and 64 more