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The density of interfaces: a new first-passage problem

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  14 July 2016

L. Chayes  and
C. Winfield
L. Chayes
Affiliation:
University of California, Los Angeles
C. Winfield*
Affiliation:
University of California, Los Angeles
*
∗ Postal address for both authors: Department of Mathematics, University of California, Los Angeles, CA 90024, USA.
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Abstract

We introduce and study a novel type of first-passage percolation problem on where the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probability p and (1 — p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e. pc > p > 1 – pc. Furthermore, we show that as p ↑ pc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

Keywords

PERCOLATIONFIRST-PASSAGE PERCOLATION

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. 82B43: Percolation 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 851 - 862
Copyright
Copyright © Applied Probability Trust 1993 

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