Random walk on hyperbolic groups and proper powers

TL;DR

This paper investigates the behavior of symmetric random walks on hyperbolic groups, showing that the probability of reaching a proper power decays exponentially at the same rate as returning to the identity element.

Contribution

It establishes a precise exponential decay rate for the probability of hitting proper powers in hyperbolic groups, linking it to return probabilities.

Findings

Proper powers are exponentially unlikely to be reached by the random walk.

Decay rate of proper power reachability matches return probability decay.

Provides new insights into the structure of hyperbolic groups and random walk behavior.

Abstract

The probability that a symmetric random walk in a hyperbolic group reaches a proper power has the same exponential rate of decay as the probability of return to the identity.