scholarly journals A phase transition in the random transposition random walk

2005 ◽  
Vol 136 (2) ◽  
pp. 203-233 ◽  
Author(s):  
Nathanaël Berestycki ◽  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Random Walk ◽  
Random Transposition

Random-walk model of the phase transition of hydrocarbon chains on a lattice

1981 ◽  
Vol 23 (2) ◽  
pp. 897-907 ◽  
Author(s):  
Sofia D. Merajver ◽  
Ellen D. Yorke ◽  
Andrew G. De Rocco
Keyword(s):  
Phase Transition ◽  
Random Walk ◽  
Random Walk Model ◽  
Hydrocarbon Chains

Phase transition in anisotropic coupled random walk

1984 ◽  
Vol 100 (6) ◽  
pp. 279-282 ◽  
Author(s):  
Horacio S. Wio ◽  
Manuel O. Cáceres
Keyword(s):  
Phase Transition ◽  
Random Walk

scholarly journals Sharpness of the Phase Transition for the Orthant Model

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
Thomas Beekenkamp
Keyword(s):  
Phase Transition ◽  
Random Walk ◽  
Critical Value ◽  
Percolation Model ◽  
Critical Threshold ◽  
Directed Percolation ◽  
Sharp Threshold ◽  
Shape Theorem ◽  
Exponentially Small

AbstractThe orthant model is a directed percolation model on $\mathbb {Z}^{d}$ ℤ d , in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

scholarly journals A phase transition in the random transposition random walk

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Nathanael Berestycki ◽  
Rick Durrett
Keyword(s):  
Phase Transition ◽  
Genome Rearrangement ◽  
Graph Model ◽  
Random Permutation ◽  
Surprising Result ◽  
Parsimony Method ◽  
Random Graph Model ◽  
Minimum Number ◽  
International Audience ◽  
Random Transposition

International audience Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.

One-Dimensional Random Walk with Phase Transition

1981 ◽  
Vol 40 (3) ◽  
pp. 485-497 ◽  
Author(s):  
O. E. Percus ◽  
J. K. Percus
Keyword(s):  
Phase Transition ◽  
Random Walk ◽  
One Dimensional

BIASED DIFFUSION: TRAP ANALYSIS IN TWO DIMENSIONS

1999 ◽  
Vol 10 (04) ◽  
pp. 753-757 ◽  
Author(s):  
ALEXANDER KIRSCH
Keyword(s):  
Phase Transition ◽  
Phase Diagram ◽  
Random Walk ◽  
Percolation Threshold ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Biased Diffusion

A method for analyzing clusters which block the random walk of particles in two-dimensional biased diffusion on percolation lattices above the percolation threshold pc is presented, focusing on the arising problems and explaining the phase transition. The difficulties in a precise trap definition are illustrated. Different trap definitions result in different trap statistics, more or less capable of capturing the trend of the phase diagram.

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