CUTOFF FOR NON-BACKTRACKING RANDOM WALKS ON SPARSE RANDOM GRAPHS

Anna Ben-Hamou (1) , Justin Salez (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Anna Ben-Hamou, Justin Salez. CUTOFF FOR NON-BACKTRACKING RANDOM WALKS ON SPARSE RANDOM GRAPHS. Annals of Probability, 2017, ⟨10.1214/16-AOP1100⟩. ⟨hal-01141192⟩