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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1207-1212 (octobre 2009)
Doi : 10.1016/j.crma.2009.08.004
Received : 6 May 2009 ;  accepted : 7 August 2009
Large deviation inequalities for supermartingales and applications to directed polymers in a random environment
Inégalités de grandes déviations pour les surmartingales et applications aux polymères dirigés en milieu aléatoire
 

Quansheng Liu a, b , Frédérique Watbled b
a Changsha University of Science and Technology, School of Mathematics and Computing Science, Changsha, 410076, PR China 
b Université de Bretagne-Sud, LMAM, campus de Tohannic, BP 573, 56017 Vannes, Université Européenne de Bretagne, France 

Abstract

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. To cite this article: Q. Liu, F. Watbled, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Résumé

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. Pour citer cet article : Q. Liu, F. Watbled, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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