Asymptotic entropy of random walks on Fuchsian buildings and Kac-Moody groups

TL;DR

This paper establishes the existence and formulas for the asymptotic entropy of isotropic random walks on regular Fuchsian buildings, linking it to the rate of escape and extending results to Kac-Moody groups.

Contribution

It proves the existence of asymptotic entropy for these walks and provides explicit formulas, generalizing previous results to a broader setting without group action assumptions.

Findings

Asymptotic entropy exists for isotropic random walks on Fuchsian buildings.

The entropy equals the rate of escape with respect to the Green distance.

Results apply to Kac-Moody groups without requiring a group action on the building.

Abstract

In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac-Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotropic random walk onto an apartment of the building, to prove existence of the asymptotic entropy for this retracted walk, and to `lift' this in order to deduce the existence of the entropy for the random walk on the building.