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|Abstract:||We study random walks on the giant component of the Erdos-Renyi random graph G(n, p) where p = lambda/n for lambda > 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 log(2) n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(log n) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (nu d)(-1) log n +/-(log n)(1/2+o(1)), where nu and d are the speed of random walk and dimension of harmonic measure on a Poisson(lambda)-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.|
|Citation:||Berestycki, Nathanael, Lubetzky, Eyal, Peres, Yuval, Sly, Allan. (2018). RANDOM WALKS ON THE RANDOM GRAPH. ANNALS OF PROBABILITY, 46 (456 - 490). doi:10.1214/17-AOP1189|
|Pages:||456 - 490|
|Type of Material:||Journal Article|
|Journal/Proceeding Title:||ANNALS OF PROBABILITY|
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