Abstract
Consider a lattice graph X realized in a k-dimensional vector space V (we shall use the same symbol X also for the set of vertices by abuse of notations). What we should have here in mind as a lattice graph is a generalization of classical lattices graphs such as the hyper-cubic lattice in ℝk, the triangular lattice and the hexagonal lattice in ℝ2. We shall show, by using our previous result [1], that, as the mesh of X goes to zero, the simple (isotropic) random walk on X “converges” to the Brownian motion on V with a suitable Euclidean structure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Kotani and T. Sunada, Albanese maps and diagonal long time asymptotics for the heat kernel, preprint 1998.
M. Kotani and T. Sunada, The Jacobian torus associated with a finite graph, preprint 1998.
M. Kotani and T. Sunada, Standard realizations of crystal lattices via harmonic maps, preprint 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Kluwer Academic Publishers
About this chapter
Cite this chapter
Kotani, M., Sunada, T. (2000). A Central Limit Theorem for the Simple Random Walk on a Crystal Lattice. In: Begehr, H.G.W., Gilbert, R.P., Kajiwara, J. (eds) Proceedings of the Second ISAAC Congress. International Society for Analysis, Applications and Computation, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0269-8_1
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0269-8_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7970-6
Online ISBN: 978-1-4613-0269-8
eBook Packages: Springer Book Archive