Critical cluster volumes in hierarchical percolation

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⁡(−β⁢∥x−y∥−d−α), where 0<α<d is fixed and β⩾0 is a parameter. We study the volume of clusters in this model at its critical point β=βc, 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 K, at criticality, namely,

In particular, we compute the critical exponent δ to be (d+α)/(d−α) when d is below the upper-critical dimension dc=3⁢α 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 d⩾3⁢α, 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⁢α, we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in ℓp∖{0} if and only if p>2⁢d/(d+α).