Long time asymptotics of non-symmetric random walks on crystal lattices
Long time asymptotics of non-symmetric random walks on crystal lattices
复制标题
晶格上非对称随机游走的长时间渐近
DOI：
10.1016/j.jfa.2016.11.011
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发表时间：
2017
影响因子：
1.7
通讯作者：
Motoko Kotani
中科院分区：
文献类型：
--
作者：
Satoshi Ishiwata;Hiroshi Kawabi;Motoko Kotani
本文从离散几何分析的角度研究了晶格上非对称随机游动的长时间渐近性。我们观察到，与晶格的标准实现相关联的欧几里德度量，称为Albanese度量，自然地出现在渐近性中。本文前半部分建立了随机游动的两类（泛函）中心极限定理。我们首先证明了具有Albanese度量的欧氏空间上的布朗运动作为随机游动的通常中心极限定理的标度极限出现。接下来，我们介绍了一个家庭的随机游动之间的原始非对称随机游动和对称化。然后，我们捕获的布朗运动的一个常数漂移的渐近方向上的欧氏空间与Albanese度量相关联的对称随机游动通过另一种中心极限定理的家庭的随机游动。本文后半部分给出了非对称随机游动的n步转移概率渐近展开式的谱几何证明。这个渐近展开是Sunada [22]，[23]得到的局部中心极限定理的一个改进，并且是[11]中关于晶格上对称随机游动的结果到非对称情形的推广。
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 then-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.