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

B. Mehrdad, S. Sen, L. Zhu

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Abstract

Abstract Consider biased random walks on two Galton–Watson trees without leaves having progeny distributions P1 and P2 (GW(P1) and GW(P2)) where P1 and P2 are supported on positive integers and P1 dominates P2 stochastically. We prove that the speed of the walk on GW(P1) is bigger than the same on GW(P2) when the bias is larger than a threshold depending on P1 and P2. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530). Résumé Nous considérons des marches aléatoires biaisées sur deux arbres de Galton–Watson sans feuilles GW(P1) et GW(P2) ayant des lois de reproduction respectivement P1 et P2, deux lois supportées par les entiers positifs telles que P1 domine stochastiquement P2. Nous prouvons que la vitesse de la marche sur GW(P1) est supérieure ou égale á celle sur GW(P2) si le biais est plus grand qu’un seuil dépendant de P1 et P2. Ceci répond partiellement á une question posée par Ben Arous, Fribergh et Sidoravicius (Comm. Pure Appl. Math. 67 (2014) 519–530).Keywords: Random walk in random environment Galton–Watson tree Speed Stochastic domination
Original languageEnglish
Pages (from-to)304-318
JournalAnnales de l'institut Henri Poincare (B): Probability and Statistics
Volume51
Issue number1
DOIs
Publication statusPublished - 2015
Externally publishedYes

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