The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias

Mehrdad, Behzad; Sen, Sanchayan; Zhu, Lingjiong

doi:10.1214/13-AIHP573

Mehrdad, Behzad ; Sen, Sanchayan ; Zhu, Lingjiong

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 304-318.

Résumé
Abstract

Nous considérons des marches aléatoires biaisées sur deux arbres de Galton–Watson sans feuilles $GW (P_{1})$ et $GW (P_{2})$ ayant des lois de reproduction respectivement $P_{1}$ et $P_{2}$ , deux lois supportées par les entiers positifs telles que $P_{1}$ domine stochastiquement $P_{2}$ . Nous prouvons que la vitesse de la marche sur $GW (P_{1})$ est supérieure ou égale á celle sur $GW (P_{2})$ si le biais est plus grand qu’un seuil dépendant de $P_{1}$ et $P_{2}$ . Ceci répond partiellement á une question posée par Ben Arous, Fribergh et Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).

Consider biased random walks on two Galton–Watson trees without leaves having progeny distributions $P_{1}$ and $P_{2}$ ( $GW (P_{1})$ and $GW (P_{2})$ ) where $P_{1}$ and $P_{2}$ are supported on positive integers and $P_{1}$ dominates $P_{2}$ stochastically. We prove that the speed of the walk on $GW (P_{1})$ is bigger than the same on $GW (P_{2})$ when the bias is larger than a threshold depending on $P_{1}$ and $P_{2}$ . This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).

MR Zbl

DOI : 10.1214/13-AIHP573

Classification : 60K37, 60J80, 60G50
Mots-clés : random walk in random environment, Galton–Watson tree, speed, stochastic domination

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     title = {The speed of a biased walk on a {Galton{\textendash}Watson} tree without leaves is monotonic with respect to progeny distributions for high values of bias},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {304--318},
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Mehrdad, Behzad; Sen, Sanchayan; Zhu, Lingjiong. The speed of a biased walk on a Galton–Watson tree without leaves is monotonic with respect to progeny distributions for high values of bias. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 304-318. doi : 10.1214/13-AIHP573. https://www.numdam.org/articles/10.1214/13-AIHP573/

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