Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Stochastic orders and the frog model

Tobias Johnson and Matthew Junge

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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 active à sa racine et un certain nombre de particules dormantes sur ses autres sites. Les particules actives suivent des chemins aléatoires, et éveillent 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 résultats étendent de nombreux théorèmes canoniques. On établit 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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 1013-1030.

Received: 29 July 2016
Revised: 16 February 2017
Accepted: 17 March 2017
First available in Project Euclid: 25 April 2018

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Frog model Transience Recurrence Phase transition Stochastic orders Increasing concave order Probability generating function order


Johnson, Tobias; Junge, Matthew. Stochastic orders and the frog model. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 1013--1030. doi:10.1214/17-AIHP830.

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