Positive definite functions and cut-off for discrete groups

TL;DR

This paper investigates the behavior of powers of positive definite functions on discrete groups, identifying conditions under which a cut-off phenomenon occurs or not, with examples from free and Coxeter groups.

Contribution

It introduces new instances of cut-off phenomena for positive definite functions on discrete groups, inspired by quantum group random walks, and explores cases without cut-off.

Findings

Cut-off occurs in free groups and infinite Coxeter groups.

Examples of absence of cut-off in free groups.

Analysis based on operator norm of the regular representation.

Abstract

We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence with respect to the operator norm of the regular representation, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.