scholarly journals On the mixing time of a simple random walk on the super critical percolation cluster

2003 ◽  
Vol 125 (3) ◽  
pp. 408-420 ◽  
Author(s):  
Itai Benjamini ◽  
Elchanan Mossel
Keyword(s):  
Random Walk ◽  
Mixing Time ◽  
Percolation Cluster ◽  
Simple Random Walk ◽  
Critical Percolation

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 CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS

2021 ◽  
pp. 1-46
Author(s):  
Charles Bordenave ◽  
Hubert Lacoin
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Mixing Time ◽  
Simple Random Walk ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Expander Graphs ◽  
Transitive Graph ◽  
Regular Tree

Abstract It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T} _d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound. It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

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.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

Random walk in a percolation cluster: external field dependence

1991 ◽  
Vol 24 (3) ◽  
pp. 735-740
Author(s):  
Jae Woo Lee ◽  
Ho Chui Kim ◽  
Jong-Jean Kim
Keyword(s):  
Random Walk ◽  
External Field ◽  
Field Dependence ◽  
Percolation Cluster

A Simple Random Walk Model for Dairy Cow Calving Time Prediction

2021 ◽  
Author(s):  
Thi Thi Zin ◽  
Pyke Tin ◽  
Pann Thinzar Seint ◽  
Kosuke Sumi ◽  
Ikuo Kobayashi ◽  
...  
Keyword(s):  
Random Walk ◽  
Dairy Cow ◽  
Simple Random Walk ◽  
Random Walk Model ◽  
Time Prediction

scholarly journals Diamond aggregation

2010 ◽  
Vol 149 (2) ◽  
pp. 351-372
Author(s):  
WOUTER KAGER ◽  
LIONEL LEVINE
Keyword(s):  
Random Walk ◽  
Growth Model ◽  
Power Law ◽  
Simple Random Walk ◽  
Internal Diffusion ◽  
Diffusion Limited Aggregation ◽  
Directional Bias ◽  
Model Based ◽  
Diffusion Limited ◽  
Logarithmic Fluctuations

AbstractInternal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

RANDOM WALK ON ANISOTROPICALLY GENERATED PERCOLATION CLUSTER AND ANISOTROPIC RANDOM WALK ON PERCOLATION CLUSTER

1989 ◽  
Vol 03 (10) ◽  
pp. 765-770
Author(s):  
C.S. KIM ◽  
MIN-HO LEE
Keyword(s):  
Random Walk ◽  
Fractal Dimensionality ◽  
Percolation Cluster ◽  
Asymptotic Convergence ◽  
Spectral Dimensionality

We studied two subjects related to anisotropy: random walk on percolation cluster having anisotropy (RWAC) and direction dependent (anisotropic) random walk on percolation cluster (AWIC). We find that the anisotropy of the cluster has only time-delaying effect on asymptotic convergence of the spectral dimensionality ds and fractal dimensionality of walk dw, however, the anisotropy of the walk results in lower spectral dimensionality and higher fractal dimensionality, as anisotropy grows larger.

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