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A Markov chain identity and monotonicity of the diffusion constants for a random walk in a heterogeneous environment

Published online by Cambridge University Press:  24 October 2008

J. B. T. M. Roerdink
J. B. T. M. Roerdink
Affiliation:
Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands
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Abstract

We consider a 2-dimensional square lattice which is partitioned into a periodic array of rectangular cells, on which a nearest neighbour random walk with symmetric increments is defined whose transition probabilities only depend on the relative position within a cell. On the basis of a determinantal identity proved in this paper, we obtain a result for finite Markov chains which shows that the diffusion constants for the random walk are monotonic functions of the individual transition probabilities. We point out the similarity of this monotonicity property to Rayleigh's Monotonicity Law for electric networks or, equivalently, reversible random walks.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 1 , July 1990 , pp. 111 - 126
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

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