Two-Dimensional Anisotropic Random Walks
A class of two-dimensional random walks on lattices is investigated in which the walker can always step to nearest neighbor sites in any row but can step to nearest neighbor sites in a column only for certain specified “connected” columns. Two cases are considered: 1) the vertical lattice “connections” occur periodically and 2) the vertical “connections” are placed randomly. In the first case the asymptotic behaviour of a number of walk properties is found to agree with the corresponding asymptotic behaviour for a walk on a translationally invariant regular two-dimensional lattice with appropriately chosen anisotropic stepping probabilities. For the case of random column connections we show that this problem is isomorphic to the problem of studying vibrations in randomly disordered harmonic lattices. The anisotropic two-dimensional lattice random walks studied in this paper may have important implications for the understanding of transport processes occurring in anisotropic crystals such as TCNQ and TTF-TCNQ and for fluid flow through packed columns.
KeywordsRandom Walk Transition Probability Matrix Random Walk Model Initial Site Dimensional Lattice
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