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On the gap between cluster dimensions of loop soups on $\mathbb{R}^3$ and the metric graph of $\mathbb{Z}^3$

Zhenhao Cai, Jian Ding

TL;DR

The paper investigates percolation of loop soups across discrete, metric-graph, and continuum settings, revealing a fundamental gap in three dimensions between the scaling limit of metric-graph loop clusters and Brownian loop-soup clusters. It establishes the existence of a one-arm exponent $\zeta\in(\tfrac{1}{2},1]$ and proves an upper bound on the upper box-counting (and thus Hausdorff) dimension of Brownian loop clusters, leading to a strict dimension gap with the metric-graph model. Through a detailed crossing-probability analysis, loop-decomposition, and a robust coupling with the Gaussian free field at critical intensity, the work shows that small-scale loops contribute meaningfully to macroscopic clusters, preventing a naive continuum limit. It also analyzes potential scaling limits in $3\le d\le 5$ and demonstrates that any such limit would exhibit incompatible one-arm and Minkowski-dimension behaviors, suggesting the scaling limit cannot equal the Brownian loop-soup cluster collection in $\mathbb{R}^3$ and highlighting the subtle role of microscale loops in shaping macroscopic fractal structure.

Abstract

The question of understanding the scaling limit of metric graph critical loop soup clusters and its relation to loop soups in the continuum appears to be one of the subtle cases that reveal interesting new scenarios about scaling limits, with mixture of macroscopic and microscopic randomness. In the present paper, we show that in three dimensions, scaling limits of the metric graph clusters are strictly larger than the clusters of the limiting continuum Brownian loop soup. We actually show that the upper box counting dimension of the latter clusters is strictly smaller than $5/2$, while that of the former is $5/2$.

On the gap between cluster dimensions of loop soups on $\mathbb{R}^3$ and the metric graph of $\mathbb{Z}^3$

TL;DR

The paper investigates percolation of loop soups across discrete, metric-graph, and continuum settings, revealing a fundamental gap in three dimensions between the scaling limit of metric-graph loop clusters and Brownian loop-soup clusters. It establishes the existence of a one-arm exponent and proves an upper bound on the upper box-counting (and thus Hausdorff) dimension of Brownian loop clusters, leading to a strict dimension gap with the metric-graph model. Through a detailed crossing-probability analysis, loop-decomposition, and a robust coupling with the Gaussian free field at critical intensity, the work shows that small-scale loops contribute meaningfully to macroscopic clusters, preventing a naive continuum limit. It also analyzes potential scaling limits in and demonstrates that any such limit would exhibit incompatible one-arm and Minkowski-dimension behaviors, suggesting the scaling limit cannot equal the Brownian loop-soup cluster collection in and highlighting the subtle role of microscale loops in shaping macroscopic fractal structure.

Abstract

The question of understanding the scaling limit of metric graph critical loop soup clusters and its relation to loop soups in the continuum appears to be one of the subtle cases that reveal interesting new scenarios about scaling limits, with mixture of macroscopic and microscopic randomness. In the present paper, we show that in three dimensions, scaling limits of the metric graph clusters are strictly larger than the clusters of the limiting continuum Brownian loop soup. We actually show that the upper box counting dimension of the latter clusters is strictly smaller than , while that of the former is .
Paper Structure (8 sections, 9 theorems, 115 equations)

This paper contains 8 sections, 9 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Crossing probability after the removal of small loops
  4. One-arm exponent and dimension of Brownian loop clusters
  5. Proof of the existence of $\zeta$
  6. Proof of $\zeta>\frac{1}{2}$
  7. Upper bound on the cluster dimension
  8. One-arm exponent and dimension of the scaling limit

Key Result

Theorem 1.2

(1) There exists $\zeta\in (\frac{1}{2},1]$ such that for any $\epsilon\in (0,1)$, (2) For any $x\in \mathbb{R}^3$ and $r_1,r_2>0$, Consequently, the upper box-counting dimension of every loop cluster of the Brownian loop soup $\mathcal{L}_{1/2}^{\mathbb{R}^3}$ is strictly less than $\frac{5}{2}$.

Theorems & Definitions (16)

  • Conjecture 1.1: werner2021clusters
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 6 more