Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T13:32:30.985Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Doyle, P. G. and Snell, J. L.. Random Walks and Electric Networks. Carus Math. Monographs no. 22 (Math. Assoc. America, 1984).CrossRefGoogle Scholar
[2]Kemeny, J. G. and Snell, J. L.. Finite Markov Chains (Van Nostrand, 1960).Google Scholar
[3]Lawler, G. F.. Weak convergence of a random walk in a random environment. Comm. Math. Phys. 87 (1982), 81–87.Google Scholar
[4]Lawler, G. F.. Low density expansion for a two-state random walk in a random environment. J. Math. Phys. 30 (1989), 145–157.Google Scholar
[5]Muir, T.. A Treatise on the Theory of Determinants (Dover, 1960).Google Scholar
[6]Roerdink, J. B. T. M. and Shuler, K. E.. Asymptotic properties of multistate random walks. I. Theory. J. Stat. Phys. 40 (1985), 205–240.Google Scholar
[7]Roerdink, J. B. T. M. and Shuler, K. E.. Asymptotic properties of multistate random walks. II. Applications to inhomogeneous periodic and random lattices. J. Stat. Phys. 41 (1985), 581–606.CrossRefGoogle Scholar
[8]Roerdink, J. B. T. M.. On the calculation of random walk properties from lattice bond enumeration. Phys. A 132 (1985), 253–268.CrossRefGoogle Scholar
[9]Roerdink, J. B. T. M., Shuler, K. E. and Lawler, G. F.. Diffusion in lattices with an-isotropic scatterers. J. Stat. Phys. 59 (1990), 23–52.CrossRefGoogle Scholar
[10]Romanovsky, V. I.. Discrete Markov Chains (Wolters-Noordhoff, 1970).Google Scholar