Random walk in a strongly inhomogeneous environment and invasion percolation
type: Item
instance of: scholarly article
title: Random walk in a strongly inhomogeneous environment and invasion percolation
zbMATH Open document ID: 0817.60097
author: Charles M. Newman
author: Daniel L. Stein
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1995-04-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77504
review text: The invasion percolation is a young branch of percolation theory. In contrast to the standard Bernoulli percolation, which in general describes the passive diffusion of fluid particles, the invasion percolation reflects properties of active displacement (for example, when one fluid forces out another one under pressure). It is shown that the invasion percolation can arise resulting from the strongly inhomogeneous limit \(\varepsilon \to 0\) of the diffusion process \[ dY_ t = \sqrt \varepsilon dB_ t - {1 \over 2} \nabla W (Y_ t)dt, \] where \(B_ t\) is the standard Wiener process and \(W\) is a (nonrandom) smooth function with countable set of local minimum points. If \(Y_ t\) is observed unfrequently [\textit{C. Kipnis} and \textit{C. M. Newman}, SIAM J. Appl. Math. 45, 972-982 (1985; Zbl 0592.60063)], it should have essentially the same asymptotic \(\varepsilon \to 0\) behaviour as a Markov jump process \(X^ \varepsilon_ t\) with the state space \(\Lambda\) consisting of points of local minima of \(W\). It is proved that if the waiting times of \(X^ \varepsilon_ t\) attribute some ordering \(\Theta\) to the graph \(\Gamma\) with the vertex set \(\Delta\) and edges between neighbouring pairs, then the sequence of transitions made for the first time coincides in the limit \(\varepsilon \to 0\) with the invasion percolation with respect to \(\Theta\). The proof is based on two useful statements concerning properties of random walks and invasion percolation, respectively.
reviewed by: A. Khorunzhy
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 82C44
Mathematics Subject Classification ID: 82B44
zbMATH DE Number: 740326
zbMATH Keywords: inhomogeneous limit
zbMATH Keywords: invasion percolation
zbMATH Keywords: Wiener process
zbMATH Keywords: random walks
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 740326