Cutoff for non-backtracking random walks on sparse random graphs

TL;DR

This paper proves the cutoff phenomenon for non-backtracking random walks on sparse random graphs, showing a sharp transition to stationarity with a universal profile under general conditions.

Contribution

It establishes the cutoff phenomenon for non-backtracking walks on sparse graphs, providing precise window and profile characterizations, a significant advance in understanding mixing times.

Findings

Cutoff occurs in non-backtracking random walks on sparse graphs.

The cutoff profile converges to a universal shape.

The results hold under general sparsity conditions.

Abstract

A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the (suitably rescaled) cutoff profile approaches a remarkably simple, universal shape.