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The density of interfaces: a new first-passage problem

Published online by Cambridge University Press:  14 July 2016

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.

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 ppc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1993 

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