Geometry of critical discrete structures: percolation on the hierarchical lattice

Consider balls $\Lambda_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and place edges independently between each pair of vertices $x\neq y\in\Lambda_n$ with probability $1-\exp(-\beta J(x, y) )$ where $J(x, y) \asymp \| x-y \|^{-\alpha}$ for some $0<\alpha<5d/6$. We identify the critical window for this model and establish the metric scaling limit within the critical window. More precisely, we show that the metric scaling limit of the maximal components is Brownian, and that this model belongs to the Erdos-Renyi universality class when $0<\alpha<5d/6$. It was recently conjectured by Hutchcroft [43, Section~7.1] that this model is a member of the Erdos-Renyi universality class when $\alpha\in(d, 4d/3]$, and we believe that this is also true for all $\alpha\in (0, d]$. This paper proves a first result in that direction. We further show that when $0<\alpha<2d/3$, the girth of each maximal component in the critical window is $\Omega_P(|\Lambda_n|^{1/3})$, contrary to the situation when $d<\alpha\leq 4d/3$ where the girth equals $3$ . Our key tool is a universality principle proven in [10, 13]. Our proof technique can be adapted for the analogous problem on long-range percolation on the discrete torus provided an upper bound on the barely subcritical two-point function is obtained.