  In the present paper, we study long time asymptotics of non-symmetric random
walks on crystal lattices from a view point of discrete geometric analysis due
to Kotani and Sunada [11, 23]. We observe that the Euclidean metric associated
with the standard realization of the crystal lattice, called the Albanese
metric, naturally appears in the asymptotics. In the former half of the present
paper, we establish two kinds of (functional) central limit theorems for random
walks. We first show that the Brownian motion on the Euclidean space with the
Albanese metric appears as the scaling limit of the usual central limit theorem
for the random walk. Next we introduce a family of random walks which
interpolates between the original non-symmetric random walk and the symmetrized
one. We then capture the Brownian motion with a constant drift of the
asymptotic direction on the Euclidean space with the Albanese metric associated
with the symmetrized random walk through another kind of central limit theorem
for the family of random walks. In the latter half of the present paper, we
give a spectral geometric proof of the asymptotic expansion of the $n$-step
transition probability for the non-symmetric random walk. This asymptotic
expansion is a refinement of the local central limit theorem obtained by Sunada
[22] and is a generalization of the result in [11] for symmetric random walks
on crystal lattices to non-symmetric cases.
