Resumen de Random walks on the random graph
Nathanaël Berestycki, Eyal Lubetzky, Yuval Peres, Allan Sly
We study random walks on the giant component of the Erdős–Rényi random graph G(n,p)G(n,p) where p=λ/np=λ/n for λ>1λ>1 fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order log2nlog2n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(logn)O(logn) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (νd)−1logn±(logn)1/2+o(1)(νd)−1logn±(logn)1/2+o(1), where νν and dd are the speed of random walk and dimension of harmonic measure on a Poisson(λ)Poisson(λ)-Galton–Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the nonbacktracking random walk.
