Simple random walk on long range percolation clusters I: heat kernel bounds

2011 ◽  
Vol 154 (3-4) ◽  
pp. 753-786 ◽  
Author(s):  
Nicholas Crawford ◽  
Allan Sly
Keyword(s):  
Random Walk ◽  
Heat Kernel ◽  
Long Range ◽  
Simple Random Walk ◽  
Percolation Clusters ◽  
Kernel Bounds

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

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

scholarly journals Erratum to “Global heat kernel bounds via desingularizing weights”

2005 ◽  
Vol 220 (1) ◽  
pp. 238-239
Author(s):  
Pierre D. Milman ◽  
Yu.A. Semenov
Keyword(s):  
Heat Kernel ◽  
Kernel Bounds
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