Large deviation and the tangent cone at infinity of a crystal lattice

  title={Large deviation and the tangent cone at infinity of a crystal lattice},
  author={M. Kotani and Toshikazu Sunada},
  journal={Mathematische Zeitschrift},
We discuss a large deviation property of a periodic random walk on a crystal lattice in view of geometry, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as we shall observe, has remarkable combinatorial features, and shows up also in the Gromov-Hausdorff limit of a crystal lattice. 

Large deviation on a covering graph with group of polynomial growth

We discuss a large deviation principle of a periodic random walk on a covering graph with its transformation group of polynomial volume growth in view of geometry. As we shall observe, the behavior

A remark on a central limit theorem for non-symmetric random walks on crystal lattices

Recently, Ishiwata, Kawabi and Kotani [4] proved two kinds of central limit theorems for non-symmetric random walks on crystal lattices from the view point of discrete geometric analysis developed by

Long time behavior of random walks on the integer lattice

  • B. Trojan
  • Mathematics
    Monatshefte für Mathematik
  • 2019
We consider an irreducible finite range random walk on the d -dimensional integer lattice and study asymptotic behavior of its transition function p ( n ;  x ) close to the boundary of Cramér’s zone.

Analysis of random walks on a hexagonal lattice

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The


Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization

Construction of continuum from a discrete surface by its iterated subdivisions

Given a trivalent graph in the 3-dimensional Euclidean space. We call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum

Hydrodynamic Limit for Weakly Asymmetric Simple Exclusion Processes in Crystal Lattices

We investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices. We construct a suitable scaling limit by using a discrete harmonic map. As we shall

Random Walks on Topological Crystals

The most exciting moment we encounter while studying mathematics is when we observe that a seemingly isolated subject turns out to be connected with other fields in an unexpected way. In the present

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

In the present paper, we study an explicit effect of non-symmetry on asymptotics of the $n$-step transition probability as $n\rightarrow \infty$ for a class of non-symmetric random walks on the



Asymptotic behavior of the transition probability of a random walk on an infinite graph

Ideas cultivated in spectral geometry are applied to obtain an asymptotic property of a reversible random walk on an infinite graph satisfying a certain periodic condition. In the course of our

Standard realizations of crystal lattices via harmonic maps

An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle

Counting geodesics which are optimal in homology

On a surface of negative curvature, we study the distribution of closed geodesics which are not far from minimizing length in their homology classes. These geodesics do not have the statistical

Jacobian Tori Associated with a Finite Graph and Its Abelian Covering Graphs

We develop a graph-theoretic analogue of the Jacobian torus, which is defined, for a finite graph X, as the torus H1(X,R)/H1(X,Z) with a natural flat metric. It is observed that this notion,


. For a certain class of reversible random walks possibly with drift on an abelian covering graph of a finite graph, using the technique of twisted transition operator, we obtain the asymptotic

Lalley's theorem on periodic orbits of hyperbolic flows

We give an asymptotic estimate for the number of periodic orbits of an Anosov flow which are subject to multi-dimensional constraints. We also study their spatial distribution. For instance, we

Albanese Maps and Off Diagonal Long Time Asymptotics for the Heat Kernel

Abstract: We discuss long time asymptotic behaviors of the heat kernel on a non-compact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact

Croissance des boules et des géodésiques fermées dans les nilvariétés

  • P. Pansu
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1983
Abstract If (M, g) is a riemannian nilmanifold, the homothetic metrics εg˜ on the universal cover M converge in the sense of Gromov for small ε. In this convergence the volume of balls and the number

Metric Structures for Riemannian and Non-Riemannian Spaces

Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-

Random Walks on Infinite Graphs and Groups

Part I. The Type Problem: 1. Basic facts 2. Recurrence and transience of infinite networks 3. Applications to random walks 4. Isoperimetric inequalities 5. Transient subtrees, and the classification