In the present paper, we study an explicit effect of nonsymmetry 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 R 2 appropriately, we observe that the Euclidean distance in 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

more »

... ain that the transition semigroup generated by the nonsymmetric random walk approximates the heat semigroup generated by the usual Brownian motion on R 2 .