Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chemical distance for smooth Gaussian fields in higher dimension

David Vernotte

TL;DR

This work studies the chemical distance in the excursion sets of smooth Gaussian fields in dimensions d >= 2, focusing on the supercritical regime ell > -ell_c. It develops a continuum percolation framework using white-noise representations to obtain finite-range approximations, and then combines deterministic local uniqueness arguments with a stochastic domination/renormalization approach to connect distant points via locally controlled paths. The main result shows that, with high probability, the chemical distance between two points is within a polylogarithmic factor of their Euclidean distance, up to a polynomial tail for deviations, thereby extending lattice Antal–Pisztora style results to continuum Gaussian fields. This advances quantitative understanding of the metric geometry of continuum percolation and lays groundwork for further analysis of level-set connectivity in high dimensions.

Abstract

Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on $\mathbb{Z}^d$. Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field $f$ and we study the topological and geometric properties of the random excursion set $\mathcal{E}_\ell(f) := \{x\in \mathbb{R}^d\ |\ f(x)\geq -\ell\}$ where $\ell\in \mathbb{R}$ is called a level. It is known that for a wide variety of fields $f$, there exists a phase transition at some critical level $\ell_c$. When $\ell> \ell_c$, the excursion set $\mathcal{E}_\ell(f)$ presents a unique unbounded component while if $\ell<\ell_c$ there are only bounded components in $\mathcal{E}_\ell(f)$. In the supercritical regime, $\ell>\ell_c$, we study the geometry of the unbounded cluster. Inspired by the work of Peter Antal and Agoston Pisztora for the Bernoulli model \cite{Antal}, we introduce the chemical distance between two points $x$ and $y$ as the Euclidean length of the shortest path connecting these points and staying in $\mathcal{E}_\ell(f)$. In this paper, we show that when $\ell>-\ell_c$ then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points.

Chemical distance for smooth Gaussian fields in higher dimension

TL;DR

This work studies the chemical distance in the excursion sets of smooth Gaussian fields in dimensions d >= 2, focusing on the supercritical regime ell > -ell_c. It develops a continuum percolation framework using white-noise representations to obtain finite-range approximations, and then combines deterministic local uniqueness arguments with a stochastic domination/renormalization approach to connect distant points via locally controlled paths. The main result shows that, with high probability, the chemical distance between two points is within a polylogarithmic factor of their Euclidean distance, up to a polynomial tail for deviations, thereby extending lattice Antal–Pisztora style results to continuum Gaussian fields. This advances quantitative understanding of the metric geometry of continuum percolation and lays groundwork for further analysis of level-set connectivity in high dimensions.

Abstract

Gaussian percolation can be seen as the generalization of standard Bernoulli percolation on . Instead of a random discrete configuration on a lattice, we consider a continuous Gaussian field and we study the topological and geometric properties of the random excursion set where is called a level. It is known that for a wide variety of fields , there exists a phase transition at some critical level . When , the excursion set presents a unique unbounded component while if there are only bounded components in . In the supercritical regime, , we study the geometry of the unbounded cluster. Inspired by the work of Peter Antal and Agoston Pisztora for the Bernoulli model \cite{Antal}, we introduce the chemical distance between two points and as the Euclidean length of the shortest path connecting these points and staying in . In this paper, we show that when then with high probability, the chemical distance between two points has a behavior close to the Euclidean distance between those two points.

Paper Structure

This paper contains 11 sections, 19 theorems, 89 equations, 2 figures.

Table of Contents

  1. General introduction and notations
  2. On the local uniqueness event
  3. Approximation of the field
  4. Deterministic argument for local uniqueness
  5. Proof of Proposition \ref{['prop:strong_perco1']}
  6. Stochastic domination argument
  7. Comparison with a Bernoulli process of high parameter
  8. Global structures
  9. Local control of the chemical distance and proof of the main theorem
  10. Local control of the chemical distance around a point
  11. Proof of Theorem \ref{['thm:theoreme_principal']}

Key Result

Theorem 1.3

In the case $d=2$, if $q$ satisfies Assumption a:a1 for some $\beta>2$ then we have $\ell_c=0$.

Figures (2)

  • Figure 1: Illustration of the event $\text{Exist}(R,\ell)$ for $d=3$. We have three different crossings (represented in different colors), each joining two opposite faces of the box.
  • Figure 2: Illustration of the proof of Proposition \ref{['prop:determinitic']}.

Theorems & Definitions (60)

  • Definition 1.1
  • Remark
  • Remark
  • Theorem 1.3: HA_critical, Threshold
  • Theorem 1.4: lcd3
  • Remark
  • Remark
  • Theorem 1.5: HA_critical,Threshold for $d=2$, severo2022uniqueness for $d\geq 3$
  • Definition 1.6
  • Remark
  • ...and 50 more