Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I

TL;DR

This paper proves central limit theorems for non-symmetric random walks on nilpotent covering graphs, linking discrete geometric analysis with Lie group theory, and extends results to functional CLTs with explicit diffusion representations.

Contribution

It establishes a semigroup CLT and a functional CLT for non-symmetric random walks on nilpotent covering graphs, using geometric realization into Lie groups and explicit diffusion process descriptions.

Findings

Semigroup CLT for non-symmetric random walks on nilpotent graphs

Functional CLT with Donsker-type invariance principle

Explicit representation of limiting diffusion process

Abstract

In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the "centered condition", we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hoelder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give several examples of random walks on nilpotent covering graphs with explicit computations.