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

2012/10/30 by Satoshi Ishiwata, Ishiwata, Satoshi, Hiroshi Kawabi +3
Mathematics · #60F05 #60G50 #60J10 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60F05 #msc:60G50 #msc:60J10

paper · pdf · doi:10.48550/arxiv.1210.7989

24pages, 4figures

arxiv created 2012/10/30 · openalex publication_date 2012/10/30 · arxiv updated 2012/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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