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Crossing exponent in the Brownian loop soup

Antoine Jego, Titus Lupu, Wei Qian

TL;DR

This work analyzes the subcritical Brownian loop soup in a bounded planar domain, focusing on one-arm crossing events across annuli and the hit behavior near fixed points. It uncovers an exact logarithmic scaling limit for single-cluster crossings, encoding it in a fixed-point function $f_\infty$ related to a 1D Brownian loop soup and a squared Bessel process, yielding the exponent $\xi_1=1-\theta$ for large-crossing probabilities. It also provides a sharp polar-set criterion in terms of $\mathrm{Cap}_{\log^\alpha}$ and reveals a second phase transition at $\theta=1$ for large-loop percolation on logarithmic scales, via a 1D/2D correspondence and regenerative-set perspectives. Together, these results connect 2D loop-soup behavior to a tractable 1D model, deliver exact asymptotics, and illuminate the percolation landscape across the subcritical regime and beyond.

Abstract

We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity $θ\in (0,1/2]$. We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii $r$ and $r^s$ as $r \to 0$ ($s >1$ fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius $r$ decays like $|\log r|^{-1+θ+ o(1)}$ as $r \to 0$. Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of $\log^α$-capacity. This paper reveals a connection between the 1D and 2D Brownian loop soups. This connection in turn implies the existence of a second critical intensity $θ= 1$ that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain.

Crossing exponent in the Brownian loop soup

TL;DR

This work analyzes the subcritical Brownian loop soup in a bounded planar domain, focusing on one-arm crossing events across annuli and the hit behavior near fixed points. It uncovers an exact logarithmic scaling limit for single-cluster crossings, encoding it in a fixed-point function related to a 1D Brownian loop soup and a squared Bessel process, yielding the exponent for large-crossing probabilities. It also provides a sharp polar-set criterion in terms of and reveals a second phase transition at for large-loop percolation on logarithmic scales, via a 1D/2D correspondence and regenerative-set perspectives. Together, these results connect 2D loop-soup behavior to a tractable 1D model, deliver exact asymptotics, and illuminate the percolation landscape across the subcritical regime and beyond.

Abstract

We study the clusters of loops in a Brownian loop soup in some bounded two-dimensional domain with subcritical intensity . We obtain an exact expression for the asymptotic probability of the existence of a cluster crossing a given annulus of radii and as ( fixed). Relying on this result, we then show that the probability for a macroscopic cluster to hit a given disc of radius decays like as . Finally, we characterise the polar sets of clusters, i.e. sets that are not hit by the closure of any cluster, in terms of -capacity. This paper reveals a connection between the 1D and 2D Brownian loop soups. This connection in turn implies the existence of a second critical intensity that describes a phase transition in the percolative behaviour of large loops on a logarithmic scale targeting an interior point of the domain.
Paper Structure (26 sections, 28 theorems, 240 equations, 1 figure)

This paper contains 26 sections, 28 theorems, 240 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof and further results
  4. Two critical values: theta = 1/2 and theta = 1
  5. Preliminaries
  6. FKG inequality in the loop soup
  7. Crossing an annulus with loops
  8. Boundary Poisson kernel in an annulus
  9. Surrounding a disk with loops
  10. A priori upper bound on the crossing probability of an annulus by a cluster
  11. Generalisation of Theorem \ref{['T:large_crossing']}
  12. One-arm event in the Brownian loop soup
  13. Tightness
  14. Integral equation
  15. Fixed-points of the integral equation and Proof of Theorem \ref{['T:convergence_crossing']}
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $\theta \in (0,1/2]$ and let $R_0 \in (0,1)$ be a macroscopic radius. Then

Figures (1)

  • Figure 3.1: Plot of the function $\alpha \mapsto c(\alpha,\theta)$ from Lemma \ref{['L:fixed-points']} for three values of $\theta$. The origin of the axes is fixed at $(0,1)$. If $\theta < 1$ (resp. $\theta > 1$), $c(\alpha,\theta)$ takes values less than 1 for some positive (resp. negative) values of $\alpha$. This reflects a phase transition at $\theta = 1$ in the uniqueness of the fixed-points of $T$\ref{['E:mapT']}, which in turn reflects a phase transition of the percolative behaviour of large loops in the loop soup (see Section \ref{['S:intro_2values']}).

Theorems & Definitions (57)

  • Theorem 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1: FKG inequality Janson84
  • ...and 47 more