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

In the present paper, we study an explicit effect of non-symmetry on asymptotics of the $n$-step transition probability as $n\to \mathrm{\infty }$ 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 ${\mathbb{R}}^2$.