Critical cluster volumes in hierarchical percolation

TL;DR

This paper analyzes the critical behavior of long-range hierarchical percolation, providing precise estimates of cluster volume distributions and revealing new scaling laws and corrections at the critical point.

Contribution

It offers the first detailed estimates of cluster volume moments and tail distributions at criticality for long-range hierarchical percolation, including new critical exponents and correction terms.

Findings

Critical exponent δ computed as (d+α)/(d−α) below the upper-critical dimension.

Precise tail estimates for cluster volume distribution at criticality.

Convergence of scaled cluster volume distribution to a chi-squared distribution in high dimensions.

Abstract

We consider long-range Bernoulli bond percolation on the $d$-dimensional hierarchical lattice in which each pair of points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $0<\alpha<d$ is fixed and $\beta \geq 0$ is a parameter. We study the volume of clusters in this model at its critical point $\beta=\beta_c$, proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up-to-constants estimates on the tail of the volume of the cluster of the origin, denoted $K$, at criticality, namely \[ \mathbb{P}_{\beta_c}(|K|\geq n) \asymp \begin{cases} n^{-(d-\alpha)/(d+\alpha)} & d < 3\alpha\\ n^{-1/2}(\log n)^{1/4} & d=3\alpha \\ n^{-1/2} & d>3\alpha. \end{cases} \] In particular, we compute the critical exponent $\delta$ to be $(d+\alpha)/(d-\alpha)$ when $d$ is below the upper-critical dimension $d_c=3\alpha$ and establish the precise order of polylogarithmic corrections to scaling at the upper-critical dimension itself. Interestingly, we find that these polylogarithmic corrections are not those predicted to hold for nearest-neighbour percolation on $\mathbb{Z}^6$ by Essam, Gaunt, and Guttmann (J. Phys. A 1978). Our work also lays the foundations for the study of the scaling limit of the model: In the high-dimensional case $d \geq 3\alpha$ we prove that the sized-biased distribution of the volume of the cluster of the origin inside a box converges under suitable normalization to a chi-squared random variable, while in the low-dimensional case $d<3\alpha$ we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in $\ell^p\setminus \{0\}$ if and only if $p>2d/(d+\alpha)$.