THE CONVERGENCE OF THE EXPLORATION PROCESS FOR CRITICAL PERCOLATION ON THE k-OUT GRAPH

Abstract

We consider the percolation on the k-out graph Gout(n, k). The critical probability of it is 1 k+ √ k2−k . Similarly to the random graph G(n, p), in a scaling window 1 k+ √ k2−k ( 1 + O(n−1/3) ) , the sequence of sizes of large components rescaled by n−2/3 converges to the excursion lengths of a Brownian motion with some drift. Also, the size of the largest component is O(log n) in the subcritical phase, and O(n) in the supercritical phase. The proof is based on the analysis of the exploration process.