Geometry of critical discrete structures: long-range percolation on the hierarchical lattice and the discrete torus
Consider (a) balls $Λ_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and (b) the $d$-dimensional discrete torus $\mathbb{T}_n^d$ on $n^d$ vertices. Place edges independently between each pair of vertices $x\neq y\inΛ_n$ or $\mathbb{T}_n^d$ with probability $1-\exp(-βJ(x, y) )$ where $J(x, y) \asymp \| x-y \|^{-α}$ for some $0<α<d$. For both of these models, we prove the following: (i) We obtain tight bounds, up to constants, on the two-point function in the barely subcritical regime. We show that in part of the barely subcritical regime, the two-point function has a plateau [47, 51]. (ii) We identify the critical window when $0<α<5d/6$. Further, using the bound on the two-point function mentioned in (i) together with a universality principle proven in [10, 14], we establish the scaling limit of the maximal components, viewed as metric measure spaces, within the critical window. More precisely, we show that the metric scaling limit of the maximal components is Brownian, and that these models belong to the Erdos-Renyi universality class when $0<α<5d/6$. It was recently conjectured by Hutchcroft [45, Section~7.1] that the model of critical hierarchical percolation with $α\in(d, 4d/3]$ is a member of the Erdos-Renyi universality class, and we believe that this is also true for all $α\in (0, d]$. Similarly, critical long-range percolation on the discrete torus is expected to be in this universality class when the effective dimension is high enough. These results take a first step in that direction. (iii) We show that when $0<α<2d/3$, the girth of each maximal component in the critical window is $Ω_P(|Λ_n|^{1/3})$ and $Ω_P(n^{d/3})$ respectively for these two models, contrary to the situation when $d<α$ where the girth would equal $3$ .