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Incipient infinite clusters and volume growth for Gaussian free fields and loop soups on metric graphs

Zhenhao Cai, Jian Ding

TL;DR

The paper investigates incipient infinite clusters for Gaussian free field level-sets and the critical loop soup on the metric graph $\widetilde{\mathbb{Z}}^d$ for all $d\ge 3$ with $d\neq 6$. It constructs four IIC types via different limiting conditionings and proves their existence and equivalence, exploiting a Basu-Sapozhinikov framework and quasi-multiplicativity from a companion work. A key finding is that the typical volume of the critical cluster under conditioning scales as $M^{(\frac{d}{2}+1)\land 4}$, supporting a self-similar scaling limit and aligning with Werner's fractal-dimension predictions. The paper also derives two-point function bounds under conditioning and establishes tightness and anti-concentration for typical volumes, underscoring the scaling behavior and stochastic nature of the IIC in these Gaussian-percolation contexts. The methodology relies on a rigorous isomorphism between loop soups and GFF, Brownian motion on metric graphs, and an analytic framework for controlling connectivity events across scales.

Abstract

In this paper, we establish the existence and equivalence of four types of incipient infinite clusters (IICs) for the critical Gaussian free field (GFF) level-set and the critical loop soup on the metric graph $\widetilde{\mathbb{Z}}^d$ for all $d\ge 3$ except the critical dimension $d=6$. These IICs are defined as four limiting conditional probabilities, involving different conditionings and various ways of taking limits: (1) conditioned on $\{0 \leftrightarrow{} \partial B(N)\}$ at criticality (where $0$ is the origin of $\mathbb{Z}^d$, and $\partial B(N)$ is the boundary of the box $B(N)$ centered at $0$ with side length $2N$), and letting $N\to \infty$; (2) conditioned on $\{0\leftrightarrow{} \infty\}$ at super-criticality, and letting the parameter tend to the critical threshold; (3) conditioned on $\{0 \leftrightarrow{} x\}$ at criticality (where $x\in \mathbb{Z}^d$ is a lattice point), and letting $x\to \infty$; (4) conditioned on the event that the capacity of the critical cluster containing $0$ exceeds $T$, and letting $T\to \infty$. Our proof employs a robust framework of Basu and Sapozhinikov (2017) for constructing IICs as in (1) and (2) for Bernoulli percolation in low dimensions (i.e., $3\le d\le 5$), where a key hypothesis on the quasi-multiplicativity is proved in our companion paper. We further show that conditioned on $\{0 \leftrightarrow{} \partial B(N)\}$, the volume of the critical cluster containing $0$ within $B(M)$ is typically of order $M^{(\frac{d}{2}+1)\land 4}$, as long as $N\gg M$. This phenomenon indicates that the critical cluster of the GFF or the loop soup exhibits self-similarity, which supports Werner's conjecture (2016) that such cluster has a scaling limit. Moreover, the exponent of $M^{(\frac{d}{2}+1)\land 4}$ matches the conjectured fractal dimension of the scaling limit proposed by Werner (2016).

Incipient infinite clusters and volume growth for Gaussian free fields and loop soups on metric graphs

TL;DR

The paper investigates incipient infinite clusters for Gaussian free field level-sets and the critical loop soup on the metric graph for all with . It constructs four IIC types via different limiting conditionings and proves their existence and equivalence, exploiting a Basu-Sapozhinikov framework and quasi-multiplicativity from a companion work. A key finding is that the typical volume of the critical cluster under conditioning scales as , supporting a self-similar scaling limit and aligning with Werner's fractal-dimension predictions. The paper also derives two-point function bounds under conditioning and establishes tightness and anti-concentration for typical volumes, underscoring the scaling behavior and stochastic nature of the IIC in these Gaussian-percolation contexts. The methodology relies on a rigorous isomorphism between loop soups and GFF, Brownian motion on metric graphs, and an analytic framework for controlling connectivity events across scales.

Abstract

In this paper, we establish the existence and equivalence of four types of incipient infinite clusters (IICs) for the critical Gaussian free field (GFF) level-set and the critical loop soup on the metric graph for all except the critical dimension . These IICs are defined as four limiting conditional probabilities, involving different conditionings and various ways of taking limits: (1) conditioned on at criticality (where is the origin of , and is the boundary of the box centered at with side length ), and letting ; (2) conditioned on at super-criticality, and letting the parameter tend to the critical threshold; (3) conditioned on at criticality (where is a lattice point), and letting ; (4) conditioned on the event that the capacity of the critical cluster containing exceeds , and letting . Our proof employs a robust framework of Basu and Sapozhinikov (2017) for constructing IICs as in (1) and (2) for Bernoulli percolation in low dimensions (i.e., ), where a key hypothesis on the quasi-multiplicativity is proved in our companion paper. We further show that conditioned on , the volume of the critical cluster containing within is typically of order , as long as . This phenomenon indicates that the critical cluster of the GFF or the loop soup exhibits self-similarity, which supports Werner's conjecture (2016) that such cluster has a scaling limit. Moreover, the exponent of matches the conjectured fractal dimension of the scaling limit proposed by Werner (2016).

Paper Structure

This paper contains 19 sections, 22 theorems, 298 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Incipient infinite cluster
  3. Volume growth
  4. Organization of the paper
  5. Preliminaries
  6. Brownian motion on $\widetilde{\mathbb{Z}}^d$
  7. Properties of the GFF
  8. Loop soup and isomorphism theorem
  9. Cluster decomposition
  10. Existence and equivalence of IICs
  11. Proof of Theorem \ref{['thm_iic']} assuming Proposition \ref{['lemma_converge_loop']}
  12. Approximation of events in IICs
  13. Proof of Proposition \ref{['lemma_converge_loop']}
  14. Two point function under conditioning of connectivity
  15. Proof of the lower bound
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $d\ge 3$ with $d\neq 6$, the limiting measures in (iic_type1), (iic_type2), (iic_type3) and (iic_type4) exist and are equivalent. Moreover, under this limiting measure, denoted by $\mathbb{P}_{d,\mathrm{IIC}}(\cdot )$, the incipient infinite cluster $\mathscr{C}^{\ge 0}$ (i.e., the cluster c

Figures (1)

  • Figure 1: In this illustration, we consider the case $\diamond=\mathbf{x}$. The two pink regions represent $\widehat{\mathcal{C}}_{K-1}$ and $\widehat{\mathcal{C}}_{K}$ respectively, where $\widehat{\mathcal{C}}_{K}$ intersects $x$. The union of the green and pink regions inside $B(n_{6K-2})$ (resp. outside $B(n_{6K})$) represents $\widehat{\mathfrak{C}}_{K-1}$ (resp. $\widehat{\mathfrak{C}}_{K}$). The two regions surrounded by dashed curves represent $\widecheck{\mathfrak{C}}_{K-1}$ and $\widecheck{\mathfrak{C}}_{K}$ respectively. Given $\widehat{\mathfrak{C}}_{K-1}$ and $\widehat{\mathfrak{C}}_{K-1}$, the dashed curves serve as absorbing boundaries for loops. The red region is a loop cluster that connects $\widehat{\mathcal{C}}_{K-1}$ and $\widehat{\mathcal{C}}_{K}$, and consists of loops disjoint from $\widecheck{\mathfrak{C}}_{K-1}\cup \widecheck{\mathfrak{C}}_{K}$.

Theorems & Definitions (38)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6: extension to the critical dimension $d=6$
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 28 more