Critical cluster volumes in hierarchical percolation

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 $p>2d/(d+\alpha)$.