Long time asymptotics of non-symmetric random walks on crystal lattices

TL;DR

This paper analyzes the long-term behavior of non-symmetric random walks on crystal lattices, revealing the role of the Albanese metric in asymptotics and establishing new central limit theorems with geometric and spectral insights.

Contribution

It introduces a geometric framework for non-symmetric random walks on crystal lattices, extending classical results to non-symmetric cases with spectral geometric proofs.

Findings

Brownian motion with Albanese metric as scaling limit

Interpolated random walks exhibit drifted Brownian motion

Asymptotic expansion refines local central limit theorem

Abstract

In the present paper, we study long time asymptotics of non-symmetric random walks on crystal lattices from a view point of discrete geometric analysis due to Kotani and Sunada [11, 23]. We observe that the Euclidean metric associated with the standard realization of the crystal lattice, called the Albanese metric, naturally appears in the asymptotics. In the former half of the present paper, we establish two kinds of (functional) central limit theorems for random walks. We first show that the Brownian motion on the Euclidean space with the Albanese metric appears as the scaling limit of the usual central limit theorem for the random walk. Next we introduce a family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We then capture the Brownian motion with a constant drift of the asymptotic direction on the Euclidean space with the Albanese metric associated with the symmetrized random walk through another kind of central limit theorem for the family of random walks. In the latter half of the present paper, we give a spectral geometric proof of the asymptotic expansion of the $n$-step transition probability for the non-symmetric random walk. This asymptotic expansion is a refinement of the local central limit theorem obtained by Sunada [22] and is a generalization of the result in [11] for symmetric random walks on crystal lattices to non-symmetric cases.