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Critical level-set percolation on finite graphs and spectral gap

Subhajit Goswami, Dipranjan Pal

TL;DR

The paper analyzes bond percolation on finite graphs induced by level-sets of the zero-average Gaussian free field on the associated metric graph. It develops a general, graph-size- and spectral-gap–driven framework via zero-average capacity and Dirichlet-form martingales to obtain mean-field critical behavior in a broad class of expanders with bounded degree. The main findings identify a critical window around $h=0$ with cluster size scaling as $|V|^{2/3}$ when $h oughly A|V|^{-1/3}$, and demarcate subcritical and supercritical regimes with precise tail bounds, showing giant components arise for $h<0$ and sublinear growth for $h>0$. The approach circumvents local tree approximations by leveraging global spectral-gap control and a Brownian embedding of an exploration martingale, with tail estimates depending only on $|V|$, degrees, and the spectral gap. These results extend mean-field phenomena to finite graphs and provide robust probabilistic controls for level-set percolation in broad graph families.

Abstract

We study the bond percolation on finite graphs induced by the level-sets of zero-average Gaussian free field on the associated metric graph above a given height (level) parameter $h \in \mathbb{R}$. We characterize the near- and off-critical phases of this model for any expanders family $\mathcal{G}_n = (V_n, E_n)$ with uniformly bounded degrees. In particular, we show that the volume of the largest open cluster at level $h_n$ is of the order $|V_n|^{\frac23}$ when $h_n$ lies in the corresponding critical window which we identify as $|h_n| = O(|V_n|^{-\frac13})$. Outside this window, the volume starts to deviate from $Θ(|V_n|^{\frac23})$ culminating into a linear order in the supercritical phase $h_n = h < 0$ (the giant component) and a logarithmic order in the subcritical phase $h_n = h > 0$. We deduce these from effective estimates on tail probabilities for the maximum volume of an open cluster at any level $h$ for a generic base graph $\mathcal{G}$. The estimates depend on $\mathcal{G}$ only through its size and upper and lower bounds on its degrees and spectral gap respectively. To the best of our knowledge, this is the first instance where a mean-field critical behavior is derived under such general setup for finite graphs. The generality of these estimates preclude any local approximation of $\mathcal{G}$ by regular infinite trees -- a standard approach in the area. Instead, our methods rely on exploiting the connection between spectral gap of the graph $\mathcal{G}$ and its connection to the level-sets of zero-average Gaussian free field mediated via a set function we call the zero-average capacity.

Critical level-set percolation on finite graphs and spectral gap

TL;DR

The paper analyzes bond percolation on finite graphs induced by level-sets of the zero-average Gaussian free field on the associated metric graph. It develops a general, graph-size- and spectral-gap–driven framework via zero-average capacity and Dirichlet-form martingales to obtain mean-field critical behavior in a broad class of expanders with bounded degree. The main findings identify a critical window around with cluster size scaling as when , and demarcate subcritical and supercritical regimes with precise tail bounds, showing giant components arise for and sublinear growth for . The approach circumvents local tree approximations by leveraging global spectral-gap control and a Brownian embedding of an exploration martingale, with tail estimates depending only on , degrees, and the spectral gap. These results extend mean-field phenomena to finite graphs and provide robust probabilistic controls for level-set percolation in broad graph families.

Abstract

We study the bond percolation on finite graphs induced by the level-sets of zero-average Gaussian free field on the associated metric graph above a given height (level) parameter . We characterize the near- and off-critical phases of this model for any expanders family with uniformly bounded degrees. In particular, we show that the volume of the largest open cluster at level is of the order when lies in the corresponding critical window which we identify as . Outside this window, the volume starts to deviate from culminating into a linear order in the supercritical phase (the giant component) and a logarithmic order in the subcritical phase . We deduce these from effective estimates on tail probabilities for the maximum volume of an open cluster at any level for a generic base graph . The estimates depend on only through its size and upper and lower bounds on its degrees and spectral gap respectively. To the best of our knowledge, this is the first instance where a mean-field critical behavior is derived under such general setup for finite graphs. The generality of these estimates preclude any local approximation of by regular infinite trees -- a standard approach in the area. Instead, our methods rely on exploiting the connection between spectral gap of the graph and its connection to the level-sets of zero-average Gaussian free field mediated via a set function we call the zero-average capacity.
Paper Structure (18 sections, 31 theorems, 212 equations)

This paper contains 18 sections, 31 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Proof outline
  5. Preliminaries
  6. The metric graph $\widetilde{\mathcal{G}}$.
  7. Excerpts from the potential theory on $\widetilde{\mathcal{G}}$.
  8. The canonical diffusions on $\widetilde{\mathcal{G}}$ and $\mathcal{G}$.
  9. The zero-average Gaussian free fields on $\mathcal{G}$ and $\widetilde{\mathcal{G}}$.
  10. zero-average capacity
  11. Martingale property of Dirichlet forms
  12. Regularity of the process $(M_K)_{K \in \mathcal{K}_{<\infty}}$
  13. Exploration of level-set components
  14. Exploration process
  15. Exploration martingale
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $d > 1$, $\lambda > 0$ and $\mathcal{G} = (V, E)$ be any finite graph with $d^\ast_\mathcal{G} \le d$ and $\lambda^\ast_\mathcal{G} \ge \lambda$. Then there exists $C = C(d, \lambda) \in (0, \infty)$ such that for any $h = A|V|^{-1/3}$ with $A \in \mathbb R$ and $\delta \in (0, 1)$ satisfying $|

Theorems & Definitions (61)

  • Theorem 1.1: Mean-field behavior inside the critical window
  • Theorem 1.2: Mean-field behavior outside the critical window
  • Theorem 1.3: Nature of phase transition on expanders
  • Lemma 2.1
  • Lemma 3.1: An expression for $f_\phi$
  • proof
  • Definition 3.2: zero-average capacity
  • Remark 3.3
  • Lemma 3.4: Variational formula for $\nu_K$
  • proof
  • ...and 51 more