Nous étudions l'entropie de la trace d'une marche aléatoire simple et symétrique de longueur n sur ℤd. Nous montrons que si d≥3, cette entropie est d'ordre n, tandis que pour d=2 elle est d'ordre n/log2n. Ces valeurs proviennent essentiellement de la taille de la frontière de la trace.
We study the entropy of the set traced by an n-step simple symmetric random walk on ℤd. We show that for d≥3, the entropy is of order n. For d=2, the entropy is of order n/log2n. These values are essentially governed by the size of the boundary of the trace.
@article{AIHPB_2010__46_4_1080_0,
author = {Benjamini, Itai and Kozma, Gady and Yadin, Ariel and Yehudayoff, Amir},
title = {Entropy of random walk range},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
volume = {46},
year = {2010},
pages = {1080-1092},
doi = {10.1214/09-AIHP345},
mrnumber = {2744887},
zbl = {1208.82046},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPB_2010__46_4_1080_0}
}
Benjamini, Itai; Kozma, Gady; Yadin, Ariel; Yehudayoff, Amir. Entropy of random walk range. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) pp. 1080-1092. doi : 10.1214/09-AIHP345. http://gdmltest.u-ga.fr/item/AIHPB_2010__46_4_1080_0/
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