Random walk on percolation clusters

1984 ◽  
Vol 29 (1) ◽  
pp. 511-514 ◽  
Author(s):  
P. Argyrakis ◽  
R. Kopelman
Keyword(s):  
Random Walk ◽  
Percolation Clusters

scholarly journals Random Walk Delayed on Percolation Clusters

2008 ◽  
Vol 45 (03) ◽  
pp. 689-702
Author(s):  
Francis Comets ◽  
François Simenhaus
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Zero Drift ◽  
Site Percolation ◽  
Dimensional Lattice ◽  
Percolation Clusters ◽  
Rate Of Escape ◽  
Weak Attraction ◽  
Large Clusters

We study a continuous-time random walk on thed-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

SELF-ATTRACTING WALK ON NON-UNIFORM SUBSTRATES

2002 ◽  
Vol 16 (12) ◽  
pp. 449-457
Author(s):  
ZHI-JIE TAN ◽  
XIAN-WU ZOU ◽  
WEI ZHANG ◽  
SHENG-YOU HUANG ◽  
ZHUN-ZHI JIN
Keyword(s):  
Monte Carlo ◽  
Random Walk ◽  
Monte Carlo Simulations ◽  
Fractal Dimensions ◽  
Attractive Interaction ◽  
Percolation Clusters ◽  
Small Scales ◽  
End Distance ◽  
The Mean ◽  
End To End

Self-attracting walk (SATW) on non-uniform substrates has been investigated by Monte Carlo simulations. The non-uniform substrates are described by Leath percolation clusters with occupied probability p. p stands for the degree of non-uniformity, and takes on values in the range pc≲p ≤1 where pc is the threshold of percolation. For the case of strong attractive interaction u, p has little influence on the walk which is dominated by attractive interactions. Furthermore, in the case of small scales, the exponent ν of the mean end-to-end distance <R2(t)> versus time t is given by ν≃1/(ds+1), while the exponent k of visited sites versus t is given by k≃ds/(ds+1), where ds are the fractal dimensions of the substrates. For u ≃ 0, the walk reduces to the random walk on percolations with p in pc≲p≤1. Also, ν and k decrease sensitively with the reduction of p. It is found, the blocked sites in the substrates (i.e. defects) have much greater influence on the walk driven by thermal flunctuation than that dominated by the attractive interaction.

scholarly journals Random Walk Delayed on Percolation Clusters

2008 ◽  
Vol 45 (3) ◽  
pp. 689-702
Author(s):  
Francis Comets ◽  
François Simenhaus
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Continuous Time Random Walk ◽  
Zero Drift ◽  
Site Percolation ◽  
Dimensional Lattice ◽  
Percolation Clusters ◽  
Rate Of Escape ◽  
Weak Attraction ◽  
Large Clusters

We study a continuous-time random walk on the d-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

scholarly journals The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Dayue Chen ◽  
Yuval Peres
Keyword(s):  
Random Walk ◽  
Percolation Cluster ◽  
Simple Random Walk ◽  
Random Perturbations ◽  
Exponential Tail ◽  
Infinite Cluster ◽  
Lamplighter Group ◽  
Percolation Clusters ◽  
Percolation Parameter ◽  
International Audience

International audience Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.

scholarly journals Return Probabilities of a Simple Random Walk on Percolation Clusters

2005 ◽  
Vol 10 (0) ◽  
pp. 250-302 ◽  
Author(s):  
Deborah Heicklen ◽  
Christopher Hoffman
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Percolation Clusters

Long-range random walk on percolation clusters

1985 ◽  
Vol 31 (9) ◽  
pp. 6008-6011 ◽  
Author(s):  
P. Argyrakis ◽  
R. Kopelman
Keyword(s):  
Random Walk ◽  
Long Range ◽  
Percolation Clusters
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