  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
$\mathbb{R}^2$.
