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

@article{Kotani2006LargeDA, title={Large deviation and the tangent cone at infinity of a crystal lattice}, author={M. Kotani and Toshikazu Sunada}, journal={Mathematische Zeitschrift}, year={2006}, volume={254}, pages={837-870} }

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.

## 38 Citations

### Large deviation on a covering graph with group of polynomial growth

- Mathematics
- 2011

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

- Mathematics
- 2017

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

- MathematicsMonatshefte 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

- Mathematics
- 2016

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…

### VORONOI TILINGS HIDDEN IN CRYSTALS —THE CASE OF MAXIMAL ABELIAN COVERINGS—

- Mathematics
- 2012

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

- Mathematics
- 2018

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…

### Random Walks on Topological Crystals

- Mathematics
- 2013

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

- Mathematics
- 2012

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…

### Random walks on graphene: generating functions, state probabilities and asymptotic behavior

- Mathematics
- 2016

We consider a discrete-time random walk on the nodes of a graphene-like graph, i.e. an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and…

## References

SHOWING 1-10 OF 19 REFERENCES

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

- Mathematics
- 1998

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

- Mathematics
- 2000

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

- MathematicsErgodic Theory and Dynamical Systems
- 2003

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

- Mathematics, Computer ScienceAdv. Appl. Math.
- 2000

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,…

### LONG TIME BEHAVIOR OF THE TRANSITION PROBABILITY OF A RANDOM WALK WITH DRIFT ON AN ABELIAN COVERING GRAPH

- Mathematics
- 2003

. 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

- MathematicsErgodic Theory and Dynamical Systems
- 1998

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

- Mathematics
- 2000

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

- MathematicsErgodic 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

- Mathematics
- 1999

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

- Mathematics
- 2000

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…