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.

Nous considérons des marches aléatoires biaisées sur deux arbres de Galton–Watson sans feuilles $\mathrm{GW}\left({P}_{1}\right)$ et $\mathrm{GW}\left({P}_{2}\right)$ 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 $\mathrm{GW}\left({P}_{1}\right)$ est supérieure ou égale á celle sur $\mathrm{GW}\left({P}_{2}\right)$ 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 ${P}_{2}$ ($\mathrm{GW}\left({P}_{1}\right)$ and $\mathrm{GW}\left({P}_{2}\right)$) 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 $\mathrm{GW}\left({P}_{1}\right)$ is bigger than the same on $\mathrm{GW}\left({P}_{2}\right)$ 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).

DOI : https://doi.org/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},
publisher = {Gauthier-Villars},
volume = {51},
number = {1},
year = {2015},
doi = {10.1214/13-AIHP573},
zbl = {06412906},
mrnumber = {3300972},
language = {en},
url = {http://www.numdam.org/articles/10.1214/13-AIHP573/}
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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. http://www.numdam.org/articles/10.1214/13-AIHP573/

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