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.

scholarly journals Gillis' Random Walks on Graphs

2005 ◽  
Vol 42 (1) ◽  
pp. 295-301 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Random Walks ◽  
Regular Graph ◽  
Simple Random Walk ◽  
Square Lattice ◽  
Random Walker ◽  
Unit Step ◽  
Random Walks On Graphs ◽  
The Green Function

We consider a random walker on a d-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of the d directions, with common probability 1/d for each one. At any later step, the random walker moves in any one of the directions, with probability q for a reversal of direction and probability p for any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is a d-dimensional square lattice. We prove that the Gillis random walk on a d-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

scholarly journals Constructing a sequence of random walks strongly converging to Brownian motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Philippe Marchal
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
International Audience

International audience We give an algorithm which constructs recursively a sequence of simple random walks on $\mathbb{Z}$ converging almost surely to a Brownian motion. One obtains by the same method conditional versions of the simple random walk converging to the excursion, the bridge, the meander or the normalized pseudobridge.

scholarly journals Gillis' Random Walks on Graphs

2005 ◽  
Vol 42 (01) ◽  
pp. 295-301 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Random Walk ◽  
Green Function ◽  
Random Walks ◽  
Regular Graph ◽  
Simple Random Walk ◽  
Square Lattice ◽  
Random Walker ◽  
Unit Step ◽  
Random Walks On Graphs ◽  
The Green Function

We consider a random walker on ad-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of theddirections, with common probability 1/dfor each one. At any later step, the random walker moves in any one of the directions, with probabilityqfor a reversal of direction and probabilitypfor any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is ad-dimensional square lattice. We prove that the Gillis random walk on ad-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the 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.

On the number of times where a simple random walk reaches its maximum

1992 ◽  
Vol 29 (02) ◽  
pp. 305-312 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Probability And Statistics

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn , are considered and their asymptotic behavior is studied.

On a Result of Aleliunas et al. Concerning Random Walks on Graphs

1990 ◽  
Vol 4 (4) ◽  
pp. 489-492 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Proof ◽  
Hitting Times ◽  
Initial State ◽  
Random Walks On Graphs ◽  
Minimum Number

Aleliunas et al. [3] proved that for a random walk on a connected raph G = (V, E) on N vertices, the expected minimum number of steps to visit all vertices is bounded by 2|E|(N - 1), regardless of the initial state. We give here a simple proof of that result through an equality involving hitting times of vertices that can be extended to an inequality for hitting times of edges, thus obtaining a bound for the expected minimum number of steps to visit all edges exactly once in each direction.

scholarly journals Type Transition of Simple Random Walks on Randomly Directed Regular Lattices

2014 ◽  
Vol 51 (4) ◽  
pp. 1065-1080 ◽  
Author(s):  
Massimo Campanino ◽  
Dimitri Petritis
Keyword(s):  
Random Walk ◽  
Periodic Function ◽  
Random Walks ◽  
Power Law ◽  
Random Perturbation ◽  
Simple Random Walk ◽  
Critical Value ◽  
Absolute Value ◽  
Horizontal Orientation ◽  
Regular Lattices

Simple random walks on a partially directed version ofZ2are considered. More precisely, vertical edges between neighbouring vertices ofZ2can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function; the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of simple random walk that is recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient.

scholarly journals NON-BACKTRACKING RANDOM WALKS MIX FASTER

2007 ◽  
Vol 09 (04) ◽  
pp. 585-603 ◽  
Author(s):  
NOGA ALON ◽  
ITAI BENJAMINI ◽  
EYAL LUBETZKY ◽  
SASHA SODIN
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Chebyshev Polynomials ◽  
Simple Random Walk ◽  
Mixing Rate ◽  
Ramanujan Graph ◽  
Mixing Rates

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than [Formula: see text] times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω( log n) times.

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