The critical density for the frog model is the degree of the tree

Johnson, Treat B.; Junge, Matthew

doi:10.1214/16-ecp29

Cited by 20 publications

(22 citation statements)

References 19 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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“…Recurrence and transience for the frog model on the d-ary tree have recently been investigated in [10] and [11] by Hoffman, Johnson and Junge. Other publications on the frog model include [2], [5], [8], [9], [12] and [13] and [18] and references therein (the list is not exhaustive).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Döbler¹,

Gantert²,

Höfelsauer³

et al. 2018

Electron. J. Probab.

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We study the frog model on Z d with drift in dimension d ≥ 2 and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension d = 2 and dimension d ≥ 3. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

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“…A sufficient condition for recurrence in this setting on Z d was given in [DP14] and improved on in [KZ16]. In our papers [HJJ16b, HJJ16a, JJ16], we prove that the frog model with simple random walk paths on the infinite d-ary tree T d switches from transient to recurrent either when the density of frogs increases with d held fixed, or when d decreases with the density held fixed.…”

Section: Introductionmentioning

confidence: 85%

Stochastic orders and the frog model

Johnson¹,

Junge²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders.This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.Le modèle de la grenouille se base sur un graphe contenant une particule activeà sa racine et un certain nombre de particules dormantes sur ses autres sites. Les particules actives suivent des chemins aléatoires, etéveillent les particules inactives qu'elles rencontrent. Nous montrons que certaines statistiques des modèles de grenouilles sont monotones en leur configuration initiale pour deux relations d'ordre: l'ordre concave croissant et l'ordre des fonctions génératrices.Ces résultatsétendent de nombreux théorèmes canoniques. Onétablit un lien entre la récurrence de la configuration initiale et la récurrence de configurations déterministes. De plus, dans le graphe naturel des entiers, la forme limite des sites activés respecte ces deux relations d'ordre. D'autres conséquences concernent des résultats de monotonicité sur la transience du modèle de grenouille où le nombre de grenouilles par site diminue lorsque l'on s'éloigne de l'origine, sur la survie dans un modèle intégrant la mort, et sur les temps de visite a un site donné dans n'importe quel modèle de grenouille.

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The critical density for the frog model is the degree of the tree

Abstract: The frog model on the rooted d-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.Comment: 12 pages; final published versio

Cited by 20 publications

References 19 publications

Sensitivity of the frog model to initial conditions

Sensitivity of the frog model to initial conditions

Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Stochastic orders and the frog model

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