Long time asymptotics of non-symmetric random walks on crystal lattices

Satoshi Ishiwata, Hiroshi Kawabi, Motoko Kotani

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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 [11,25]. 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 [22,23] and is a generalization of the result in [11] for symmetric random walks on crystal lattices to non-symmetric cases.

Original languageEnglish
Pages (from-to)1553-1624
Number of pages72
JournalJournal of Functional Analysis
Volume272
Issue number4
DOIs
Publication statusPublished - Feb 15 2017

Fingerprint

Long-time Asymptotics
Random walk
Crystal
Central limit theorem
Metric
Brownian motion
Asymptotic Expansion
Euclidean space
Functional Central Limit Theorem
Geometric proof
Geometric Analysis
Scaling Limit
Transition Probability
Euclidean
Refinement
Interpolate

Keywords

  • Asymptotic expansion
  • Central limit theorem
  • Crystal lattice
  • Non-symmetric random walk

ASJC Scopus subject areas

  • Analysis

Cite this

Long time asymptotics of non-symmetric random walks on crystal lattices. / Ishiwata, Satoshi; Kawabi, Hiroshi; Kotani, Motoko.

In: Journal of Functional Analysis, Vol. 272, No. 4, 15.02.2017, p. 1553-1624.

Research output: Contribution to journalArticle

Ishiwata, Satoshi ; Kawabi, Hiroshi ; Kotani, Motoko. / Long time asymptotics of non-symmetric random walks on crystal lattices. In: Journal of Functional Analysis. 2017 ; Vol. 272, No. 4. pp. 1553-1624.
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