Domany-Kinzel Model of Directed Percolation: Formulation as a Random-Walk Problem and Some Exact Results

1982 ◽  
Vol 48 (12) ◽  
pp. 775-778 ◽  
Author(s):  
F. Y. Wu ◽  
H. Eugene Stanley
Keyword(s):  
Random Walk ◽  
Directed Percolation ◽  
Exact Results

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

scholarly journals Exact results for the roughness of a finite size random walk

2006 ◽  
Vol 370 (1) ◽  
pp. 127-131 ◽  
Author(s):  
V. Alfi ◽  
F. Coccetti ◽  
M. Marotta ◽  
A. Petri ◽  
L. Pietronero
Keyword(s):  
Random Walk ◽  
Finite Size ◽  
Exact Results

scholarly journals Some approximate results in renewal and dam theories

1971 ◽  
Vol 12 (4) ◽  
pp. 425-432 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walk ◽  
Sequential Analysis ◽  
Random Variables ◽  
Renewal Theory ◽  
Slight Modification ◽  
Exact Results ◽  
Fundamental Identity ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Two Barriers

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

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 Directed Percolation and Random Walk

2002 ◽  
pp. 273-297 ◽  
Author(s):  
Geoffrey Grimmett ◽  
Philipp Hiemer
Keyword(s):  
Random Walk ◽  
Directed Percolation

From rigid rod to random walk. Some exact results

1993 ◽  
Vol 30 (5) ◽  
pp. 617-620 ◽  
Author(s):  
J. Rieger
Keyword(s):  
Random Walk ◽  
Exact Results ◽  
Rigid Rod

Elements of the Random Walk

2004 ◽  
Author(s):  
Joseph Rudnick ◽  
George Gaspari
Keyword(s):  
Random Walk
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