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

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In the present paper, we study an explicit effect of non-symmetry on asymptotics of the n-step transition probability as n→ ∞ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular lattice into R² appropriately, we observe that the Euclidean distance in R² 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 R².