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

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 304-318 / Harvested from Numdam

Résumé

Nous considérons des marches aléatoires biaisées sur deux arbres de Galton–Watson sans feuilles $\mathrm {GW}(P_{1})$ et $\mathrm {GW}(P_{2})$ 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 $\mathrm {GW}(P_{1})$ est supérieure ou égale á celle sur $\mathrm {GW}(P_{2})$ 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).

Abstract

Consider biased random walks on two Galton–Watson trees without leaves having progeny distributions $P_{1}$ and $P_{2}$ ($\mathrm {GW}(P_{1})$ and $\mathrm {GW}(P_{2})$) 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 $\mathrm {GW}(P_{1})$ is bigger than the same on $\mathrm {GW}(P_{2})$ 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).

Détail

Publié le : 2015-01-01

MR 3300972 | Zbl 06412906

DOI : https://doi.org/10.1214/13-AIHP573
Classification:  60K37,  60J80,  60G50

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