From transience to recurrence with Poisson tree frogs

Hoffman, Christopher; Johnson, Treat B.; Junge, Matthew

doi:10.1214/15-aap1127

Cited by 42 publications

(79 citation statements)

References 28 publications

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“…On the other hand, the frog model on the infinite d-ary tree can be either transient a.s. or recurrent a.s., depending on the initial configuration. For example, on the d-ary tree when (η(v)) v is i.i.d.-Poisson(µ), the frog model is recurrent or transient depending on whether µ is greater or less than a critical value µ c (d) [5,7]. In [8], the authors give a theorem comparing frog models with different initial conditions on the same graph, which shows that the frog model on the d-ary tree is recurrent if η(v) pgf Poisson(µ) for all v for some µ > µ c (d).…”

Section: Introductionmentioning

confidence: 99%

Sensitivity of the frog model to initial conditions

Johnson

Rolla

2019

Electron. Commun. Probab.

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The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.

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Section: Introductionmentioning

confidence: 99%

Sensitivity of the frog model to initial conditions

Johnson

Rolla

2019

Electron. Commun. Probab.

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“…We study two problems presented to us by I. Benjamini [8] (see also [20,Open Question 5]). In these two problems we seek an estimate which holds uniformly in the identity of the origin o. Deterministically, CT(T d,n ) ≥ n. Conversely, CT(T d,n ) can be bounded from above by the cover time of T d,n by a single SRW, which Aldous [1] showed is ≍ n 2 d n log d. To the best of the author's knowledge, the best previously known upper bound on CT(T d,n ) is exponential in n, i.e.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Frogs on trees?

Hermon¹

2018

Electron. J. Probab.

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We study a system of simple random walks on T d,n = (V d,n , E d,n ), the d-ary tree of depth n, known as the frog model. Initially there are Pois(λ) particles at each site, independently, with one additional particle planted at some vertex o. Initially all particles are inactive, except for the ones which are placed at o. Active particles perform independent simple random walk on the tree of length t ∈ N ∪ {∞}, referred to as the particles' lifetime. When an active particle hits an inactive particle, the latter becomes active. The model is often interpreted as a model for a spread of an epidemic. As such, it is natural to investigate whether the entire population is eventually infected, and if so, how quickly does this happen. Let R t be the set of vertices which are visited by the process (with lifetime t). The susceptibility S(T d,n ) := inf{t : R t = V d,n } is the minimal lifetime required for the process to visit all sites. The cover time CT(T d,n ) is the first time by which every vertex was visited at least once, when we take t = ∞. We show that there exist absolute constants c, C > 0 such that for all d ≥ 2 and all λ = λ n > 0 which does not diverge nor vanish too rapidly as a function of n, with high probability c ≤ λS(T d,n )/[n log(n/λ)] ≤ C and CT(T d,n ) ≤ 3 4 √ log |V d,n | . . Financial support by the EPSRC grant EP/L018896/1. 1 In the year following the time at which the first draft of this paper was posted online, two other papers concerning the frog model on finite graphs were posted online. The first is [9], which is some sense a continuation of this work. The base graphs considered in [9] are (1) d-dim tori of side length n for all d ≥ 1 and (2) expanders. The second is [21] which contains some related results to our main results, which we shall discuss in more detail.

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“…Next we compute E[x V ; D i ] for each i beginning with i = 1. represents having all frogs that go to a from the sub-tree rooted at b then travel to the root; and D (2) 2 represents having at least one frog that travels to a from the sub-tree rooted at b then go to b ′ (i.e. D 2 /D…”

Section: Constructing the Operatormentioning

confidence: 99%

Recurrence of the frog model on the 3,2-alternating tree

Rosenberg¹

2018

ALEA

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Abstract. Consider a growing system of random walks on the 3,2-alternating tree, where generations of nodes alternate between having two and three children. Any time a particle lands on a node which has not been visited previously, a new particle is activated at that node, and begins its own random walk. The model described belongs to a class of problems that are collectively referred to as the frog model. Building on a recent proof of recurrence (meaning infinitely many frogs hit the root with probability one) on the regular binary tree, this paper establishes recurrence for the 3,2-alternating case.

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From transience to recurrence with Poisson tree frogs

Cited by 42 publications

References 28 publications

Sensitivity of the frog model to initial conditions

Sensitivity of the frog model to initial conditions

Frogs on trees?

Recurrence of the frog model on the 3,2-alternating tree

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