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.
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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.
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© 2000 Kluwer Academic Publishers
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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
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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
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