Critical Long Range Percolation: Scaling Limits for Small β

Abstract

A long-range percolation (LRP) graph has the integers as vertices and an edge between every pair of vertices x and y with probability β(x − y) −s for some positive parameters β and s. These graphs are well-understood for s 6= 2. In the case of s = 2, far less is known. Ding and Sly [4] raised the open question of whether there is a joint scaling of LRP graphs to metric spaces and random walks on LRP graphs to diffusion processes for s = 2. We make some progress on this problem by using multi-scale analysis to prove the existence of a scaling limit for continuous LRP graphs as random metric spaces in the regime of very small β. In the future, we hope to use this as a starting point to address random walks on these graphs and larger values of β.