### 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 language | English |
---|---|

Pages (from-to) | 1553-1624 |

Number of pages | 72 |

Journal | Journal of Functional Analysis |

Volume | 272 |

Issue number | 4 |

DOIs | |

Publication status | Published - Feb 15 2017 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*272*(4), 1553-1624. https://doi.org/10.1016/j.jfa.2016.11.011

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

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 272, no. 4, pp. 1553-1624. https://doi.org/10.1016/j.jfa.2016.11.011

}

TY - JOUR

T1 - Long time asymptotics of non-symmetric random walks on crystal lattices

AU - Ishiwata, Satoshi

AU - Kawabi, Hiroshi

AU - Kotani, Motoko

PY - 2017/2/15

Y1 - 2017/2/15

N2 - 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.

AB - 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.

KW - Asymptotic expansion

KW - Central limit theorem

KW - Crystal lattice

KW - Non-symmetric random walk

UR - http://www.scopus.com/inward/record.url?scp=85007524020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85007524020&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2016.11.011

DO - 10.1016/j.jfa.2016.11.011

M3 - Article

VL - 272

SP - 1553

EP - 1624

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 4

ER -