Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large deviations for partition functions of directed polymers in an IID field

Iddo Ben-Ari

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Consider the partition function of a directed polymer in ℤd, d≥1, in an IID field. We assume that both tails of the negative and the positive part of the field are at least as light as exponential. It is well known that the free energy of the polymer is equal to a deterministic constant for almost every realization of the field and that the upper tail of the large deviations is exponential. The lower tail of the large deviations is typically lighter than exponential. In this paper we obtain sharp estimates on the lower tail of the large deviations given in terms of the distribution of the IID field. Our proofs are also applicable to the model of directed last passage percolation and (non-directed) first passage percolation.


Considérons la fonction de partition d’un polymère dirigé dans ℤd, d≥1, d’un champ IID. On suppose que les queues des parties positive et négative sont au moins aussi légères qu’une exponentielle. Il est bien connu que l’énergie libre du polymère est égale à une constante déterministe pour presque toute réalisation du champ et que la queue supérieure des grandes déviations est exponentielle. La queue inférieure des grandes déviations est typiquement plus légère qu’une exponentielle. Dans cet article nous obtenons des estimations précises sur la queue inférieure des grandes déviations en fonction de la distribution du champ IID. Nos preuves sont également applicables au modèle de percolation de dernier passage dirigé et à celui de percolation de premier passage (non dirigé).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 3 (2009), 770-792.

First available in Project Euclid: 4 August 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F10: Large deviations

Large deviations Partition function Last passage percolation


Ben-Ari, Iddo. Large deviations for partition functions of directed polymers in an IID field. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 3, 770--792. doi:10.1214/08-AIHP185.

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