scholarly journals Long Term Behaviour of a Reversible System of Interacting Random Walks

2019 ◽  
Vol 175 (1) ◽  
pp. 71-96 ◽  
Author(s):  
Svante Janson ◽  
Vadim Shcherbakov ◽  
Stanislav Volkov
Keyword(s):  
Random Walks ◽  
Reversible System ◽  
Interacting Random Walks

Out-of-equilibrium random walks

2020 ◽  
Vol 52 (3) ◽  
pp. 772-797
Author(s):  
Leonardo A. Videla
Keyword(s):  
Stochastic Process ◽  
Random Walks ◽  
Complete Graphs ◽  
Knowledge Process ◽  
Random Walker ◽  
Underlying Graph ◽  
Graph Sequence

AbstractWe study the long-term behaviour of a random walker embedded in a growing sequence of graphs. We define a (generally non-Markovian) real-valued stochastic process, called the knowledge process, that represents the ratio between the number of vertices already visited by the walker and the current size of the graph. We mainly focus on the case where the underlying graph sequence is the growing sequence of complete graphs.

Non-Classical Interacting Random Walks

2007 ◽  
pp. 1521-1574
Author(s):  
Francis Comets ◽  
Martin Zerner
Keyword(s):  
Random Walks ◽  
Interacting Random Walks

scholarly journals Frogs and some other interacting random walks models

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Serguei Yu. Popov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Active Particle ◽  
Open Problems ◽  
Simple Version ◽  
Random Walks On Graphs ◽  
Interacting Random Walks ◽  
International Audience ◽  
Frog Model

International audience We review some recent results for a system of simple random walks on graphs, known as \emphfrog model. Also, we discuss several modifications of this model, and present a few open problems. A simple version of the frog model can be described as follows: There are active and sleeping particles living on some graph. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active.

On the range of the transient frog model on ℤ

2017 ◽  
Vol 49 (2) ◽  
pp. 327-343 ◽  
Author(s):  
Arka Ghosh ◽  
Steven Noren ◽  
Alexander Roitershtein
Keyword(s):  
Random Walk ◽  
Lower Bound ◽  
Random Walks ◽  
Explicit Formula ◽  
Infinite System ◽  
Scaling Limits ◽  
Interacting Random Walks ◽  
Degenerate Distribution ◽  
Frog Model ◽  
Asymptotic Scaling

Abstract We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

The nonhomogeneous frog model on ℤ

2018 ◽  
Vol 55 (4) ◽  
pp. 1093-1112
Author(s):  
Josh Rosenberg
Keyword(s):  
Positive Integer ◽  
Random Walks ◽  
Stochastic Order ◽  
Random Variables ◽  
Independent Random Variables ◽  
Active Particle ◽  
Integer Points ◽  
Interacting Random Walks ◽  
Frog Model ◽  
Positive Integers

Abstract We examine a system of interacting random walks with leftward drift on ℤ, which begins with a single active particle at the origin and some distribution of inactive particles on the positive integers. Inactive particles become activated when landed on by other particles, and all particles beginning at the same point possess equal leftward drift. Once activated, the trajectories of distinct particles are independent. This system belongs to a broader class of problems involving interacting random walks on rooted graphs, referred to collectively as the frog model. Additional conditions that we impose on our model include that the number of frogs (i.e. particles) at positive integer points is a sequence of independent random variables which is increasing in terms of the standard stochastic order, and that the sequence of leftward drifts associated with frogs originating at these points is decreasing. Our results include sharp conditions with respect to the sequence of random variables and the sequence of drifts that determine whether the model is transient (meaning the probability infinitely many frogs return to the origin is 0) or nontransient. We consider several, more specific, versions of the model described, and a cleaner, more simplified set of sharp conditions will be established for each case.

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