An explicit effect of non-symmetry of random walks on the triangular lattice

TL;DR

This paper investigates how non-symmetry influences the long-term behavior of certain random walks on a triangular lattice, revealing geometric and probabilistic connections to heat diffusion and Brownian motion.

Contribution

It explicitly characterizes the asymptotic behavior of non-symmetric random walks on the triangular lattice and links it to Euclidean geometry and heat semigroup approximation.

Findings

Euclidean distance appears in asymptotics

Transition semigroup approximates heat semigroup

Characterization of lattice realization from a geometric viewpoint

Abstract

In the present paper, we study an explicit effect of non-symmetry on asymptotics of the $n$-step transition probability as $n\rightarrow \infty$ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular lattice into $\mathbb{R}^2$ appropriately, we observe that the Euclidean distance in $\mathbb{R}^2$ naturally appears in the asymptotics. We characterize this realization from a geometric view point of Kotani-Sunada's standard realization of crystal lattices. As a corollary of the main theorem, we prove that the transition semigroup generated by the non-symmetric random walk approximates the heat semigroup generated by the usual Brownian motion on $\mathbb{R}^2$.