Critical cluster volumes in hierarchical percolation

Abstract

We consider long‐range Bernoulli bond percolation on the ‐dimensional hierarchical lattice in which each pair of points and are connected by an edge with probability , where is fixed and is a parameter. We study the volume of clusters in this model at its critical point , 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 as , at criticality, namely, In particular, we compute the critical exponent to be when is below the upper‐critical dimension and establish the precise order of polylogarithmic corrections to scaling at the upper‐critical dimension itself. Our work also lays the foundations for the study of the scaling limit of the model: In the high‐dimensional case , 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 , we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in if and only if .