Large deviations for partition functions of directed polymers in an IID field
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 770-792.

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).

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.

DOI : 10.1214/08-AIHP185
Classification : 60K35, 60F10
Mots-clés : large deviations, partition function, last passage percolation 
@article{AIHPB_2009__45_3_770_0,
     author = {Ben-Ari, Iddo},
     title = {Large deviations for partition functions of directed polymers in an {IID} field},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {770--792},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     doi = {10.1214/08-AIHP185},
     mrnumber = {2548503},
     zbl = {1176.60080},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP185/}
}
TY  - JOUR
AU  - Ben-Ari, Iddo
TI  - Large deviations for partition functions of directed polymers in an IID field
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 770
EP  - 792
VL  - 45
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP185/
DO  - 10.1214/08-AIHP185
LA  - en
ID  - AIHPB_2009__45_3_770_0
ER  - 
%0 Journal Article
%A Ben-Ari, Iddo
%T Large deviations for partition functions of directed polymers in an IID field
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 770-792
%V 45
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP185/
%R 10.1214/08-AIHP185
%G en
%F AIHPB_2009__45_3_770_0
Ben-Ari, Iddo. Large deviations for partition functions of directed polymers in an IID field. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 770-792. doi : 10.1214/08-AIHP185. http://www.numdam.org/articles/10.1214/08-AIHP185/

[1] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl

[2] P. Carmona and Y. Hu. On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 (2002) 431-457. | MR | Zbl

[3] Y. Chow and Y. Zhang. Large deviations in first-passage percolation. Ann. Appl. Probab. 13 (2003) 1601-1614. | MR | Zbl

[4] F. Comets, T. Shiga and N. Yoshida. Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 (2003) 705-723. | MR | Zbl

[5] M. Cranston, D. Gauthier and T. S. Mountford. On large deviations regimes for random media models. Ann. Appl. Probab. 19 (2009) 826-862. | MR | Zbl

[6] M. Cranston, T. S. Mountford and T. Shiga. Lyapunov exponents for the parabolic Anderson model. Acta Math. Univ. Comenian. (N.S.) 71 (2002) 163-188. | MR | Zbl

[7] J.-D. Deuschel and O. Zeitouni. On increasing subsequences of I.I.D. samples. Combin. Probab. Comput. 8 (1999) 247-263. | MR | Zbl

[8] H. Kesten. Aspects of first passage percolation. In École d'été de probabilités de Saint-Flour, XIV-1984 125-264. Lecture Notes in Math. 1180. Springer, Berlin, 1986. | MR | Zbl

[9] T. Seppäläinen. Large deviations for increasing sequences on the plane. Probab. Theory Related Fields 112 (1998) 221-244. | MR | Zbl

Cité par Sources :