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

@article{AIHPB_2015__51_1_304_0,
     author = {Mehrdad, Behzad and Sen, Sanchayan and Zhu, Lingjiong},
     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},
     mrnumber = {3300972},
     zbl = {1314.60160},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/13-AIHP573/}
}

TY  - JOUR
AU  - Mehrdad, Behzad
AU  - Sen, Sanchayan
AU  - Zhu, Lingjiong
TI  - 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
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 304
EP  - 318
VL  - 51
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/13-AIHP573/
DO  - 10.1214/13-AIHP573
LA  - en
ID  - AIHPB_2015__51_1_304_0
ER  -

%0 Journal Article
%A Mehrdad, Behzad
%A Sen, Sanchayan
%A Zhu, Lingjiong
%T 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
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 304-318
%V 51
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/13-AIHP573/
%R 10.1214/13-AIHP573
%G en
%F AIHPB_2015__51_1_304_0

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/

Bibliographie
Cité par

[1] E. Aïdékon. Speed of the biased random walk on a Galton–Watson tree. Probab. Theory Related Fields. To appear, 2014. DOI:10.1007/s00440-013-0515-y. | DOI | MR | Zbl

[2] K. B. Athreya. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab. 4 (1994) 779–790. | MR | Zbl

[3] G. Ben Arous, A. Fribergh and V. Sidoravicius. Lyons–Pemantle–Peres monotonicity problem for high biases. Comm Pure Appl. Math. 67 (2014) 519–530. | DOI | MR | Zbl

[4] N. H. Bingham. On the limit of a supercritical branching process. J. Appl. Probab. 25 (1988) 215–228. | DOI | MR | Zbl

[5] H. Kesten and B. P. Stigum. A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37 (1966) 1211–1223. | DOI | MR | Zbl

[6] R. Lyons. Random walks and percolation on trees. Ann. Probab. 18 (1990) 931–958. | DOI | MR | Zbl

[7] R. Lyons, R. Pemantle and Y. Peres. Ergodic theory on Galton–Watson trees: Speed of random walk and dimension of harmonic measure. Erg. Theory Dynam. Syst. 15 (1995) 593–619. | DOI | MR | Zbl

[8] R. Lyons, R. Pemantle and Y. Peres. Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106 (1996) 254–268. | DOI | MR | Zbl

[9] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of $L log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125–1138. | DOI | MR | Zbl

[10] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin, 2004. | MR | Zbl

Cité par Sources :