[Inégalités de grandes déviations pour les surmartingales et applications aux polymères dirigés en milieu aléatoire]
Nous montrons d'abord des inégalités exponentielles de grandes déviations pour les surmartingales, qui sont optimales dans le cas de variables aléatoires indépendantes et identiquement distribuées (iid). Nous utilisons ensuite ce résultat pour établir des inégalités exponentielles de concentration pour l'énergie libre des polymères dirigés en milieu aléatoire ; nous en déduisons des bornes supérieures pour sa vitesse de convergence (en probabilité, presque sûrement, et dans ), et une expression de l'énergie libre en terme d'énergies libres de certaines cascades multiplicatives.
We first show exponential large deviation inequalities for supermartingales, which are optimal in the independent and identically distributed (iid) case. We then apply them to establish new exponential concentration inequalities for the free energy of directed polymers in random environment; as consequences we obtain upper bounds of its rate of convergences (in probability, almost surely, and in ), and give an expression for the free energy in terms of free energies of some multiplicative cascades.
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@article{CRMATH_2009__347_19-20_1207_0,
author = {Quansheng Liu and Fr\'ed\'erique Watbled},
title = {Large deviation inequalities for supermartingales and applications to directed polymers in a random environment},
journal = {Comptes Rendus. Math\'ematique},
pages = {1207--1212},
publisher = {Elsevier},
volume = {347},
number = {19-20},
year = {2009},
doi = {10.1016/j.crma.2009.08.004},
language = {en},
}
TY - JOUR AU - Quansheng Liu AU - Frédérique Watbled TI - Large deviation inequalities for supermartingales and applications to directed polymers in a random environment JO - Comptes Rendus. Mathématique PY - 2009 SP - 1207 EP - 1212 VL - 347 IS - 19-20 PB - Elsevier DO - 10.1016/j.crma.2009.08.004 LA - en ID - CRMATH_2009__347_19-20_1207_0 ER -
%0 Journal Article %A Quansheng Liu %A Frédérique Watbled %T Large deviation inequalities for supermartingales and applications to directed polymers in a random environment %J Comptes Rendus. Mathématique %D 2009 %P 1207-1212 %V 347 %N 19-20 %I Elsevier %R 10.1016/j.crma.2009.08.004 %G en %F CRMATH_2009__347_19-20_1207_0
Quansheng Liu; Frédérique Watbled. Large deviation inequalities for supermartingales and applications to directed polymers in a random environment. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1207-1212. doi : 10.1016/j.crma.2009.08.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.08.004/
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