[Paper Review] Limit theorems for random walks on Fuchsian buildings and Kac-Moody groups

Lorenz A. Gilch, Sebastian Mueller+1 more|arXiv (Cornell University)|Nov 27, 2014

Geometric and Algebraic Topology24 references3 citations

TL;DR

This paper establishes a rate of escape and central limit theorem for isotropic random walks on Fuchsian buildings and associated Kac-Moody groups, using a renewal structure built via cone types and strongly connected automata. The key contribution is explicit formulae for the speed and asymptotic variance, extending planar hyperbolic results to non-planar, higher-rank buildings via retraction techniques and automaton-based analysis.

ABSTRACT

In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac-Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in [21]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building.

Motivation & Objective

• To extend central limit theorems from hyperbolic and Lie group settings to Fuchsian buildings and Kac-Moody groups.
• To establish a rate of escape and central limit theorem for isotropic random walks on buildings of non-affine, non-spherical type.
• To develop a renewal structure for random walks on buildings using cone types and automata.
• To prove strong connectivity of the associated Cannon automaton for various Coxeter group types.
• To control moments of renewal times via retraction onto apartments, enabling limit theorems.

Proposed method

• Define cones and cone types in Fuchsian buildings to construct a renewal process.
• Prove that the associated automaton on the Coxeter group is strongly connected for all but a few exceptional cases.
• Use retraction of random walks onto apartments to control moments of renewal times.
• Adapt techniques from Haissinski, Mathieu, and Müller (2013) for planar surface groups to non-planar buildings.
• Employ algebraic and geometric properties of Coxeter groups and buildings to analyze cone type transitions.
• Verify strong connectivity of the automaton across four classes of triangle Coxeter groups via path constructions.

Experimental results

Research questions

• RQ1 Can a central limit theorem be established for random walks on Fuchsian buildings beyond the planar or hyperbolic setting?
• RQ2 What conditions on the Coxeter group ensure that the cone-type automaton is strongly connected?
• RQ3 How can the speed and asymptotic variance of isotropic random walks on buildings be explicitly computed?
• RQ4 To what extent can renewal theory techniques from surface groups be generalized to buildings of higher rank?
• RQ5 What role does the retraction onto an apartment play in controlling the moments of renewal times?

Key findings

• A rate of escape theorem is proven, providing an explicit formula for the speed of isotropic random walks on Fuchsian buildings.
• A central limit theorem is established, with a formula for the asymptotic variance of the walk.
• The Cannon automaton associated with the building’s cone types is strongly connected for all but finitely many Coxeter groups of triangle type.
• The renewal structure is successfully constructed via cone types, enabling the application of limit theorem techniques.
• The retraction of walks onto apartments allows effective control of the moments of renewal times.
• The results apply not only to buildings arising from Kac-Moody groups but also to general Fuchsian buildings without a group structure.