Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - 1990 - Volume 108, Issue 01  

Research Article

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

J. B. T. M. Roerdinka1

a1 Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands

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.

(Received October 05 1989)

(Revised November 09 1989)