Scaling limit of dynamical percolation on critical Erdős–Rényi random graphs

Critical percolation on random regular graphs

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d is fixed and n grows. We prove mean-field behavior around the critical probability pc = 1 d−1 . In particular, we show that there is a scaling window of width n around pc in which the sizes of the largest components are roughly n 2/3 and we describe their limiting joint ...

Scaling limit of simple random walk on supercritical percolation clusters

The Scaling Limit Geometry of Near-Critical 2D Percolation

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc + λδ 1/ν , with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (crit...

On the Critical Density for Percolation in Random Geometric Graphs

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density λ c for d-dimensional Poisson random geometric graphs in continuum percolation theory. In two-dimensional space with the Euclidean norm, simulation studies show λ c ≈ 1.44, while the best theoretical bounds obtained thus far...

The continuum limit of critical random graphs

We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n+ λn−4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n−1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the t...