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

Iddo Ben-Ari

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 770-792
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Ben-Ari, Iddo. "Large deviations for partition functions of directed polymers in an IID field." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 770-792. <http://eudml.org/doc/78043>.

@article{Ben2009,
abstract = {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.},
author = {Ben-Ari, Iddo},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {large deviations; partition function; last passage percolation},
language = {eng},
number = {3},
pages = {770-792},
publisher = {Gauthier-Villars},
title = {Large deviations for partition functions of directed polymers in an IID field},
url = {http://eudml.org/doc/78043},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 770
EP - 792
AB - 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.
LA - eng
KW - large deviations; partition function; last passage percolation
UR - http://eudml.org/doc/78043
ER -

References

top
  1. [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. Zbl0617.26001MR898871
  2. [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. Zbl1015.60100MR1939654
  3. [3] Y. Chow and Y. Zhang. Large deviations in first-passage percolation. Ann. Appl. Probab. 13 (2003) 1601–1614. Zbl1038.60093MR2023891
  4. [4] F. Comets, T. Shiga and N. Yoshida. Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 (2003) 705–723. Zbl1042.60069MR1996276
  5. [5] M. Cranston, D. Gauthier and T. S. Mountford. On large deviations regimes for random media models. Ann. Appl. Probab. 19 (2009) 826–862. Zbl1171.60023MR2521889
  6. [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. Zbl1046.60057MR1980378
  7. [7] J.-D. Deuschel and O. Zeitouni. On increasing subsequences of I.I.D. samples. Combin. Probab. Comput. 8 (1999) 247–263. Zbl0949.60019MR1702546
  8. [8] H. Kesten. Aspects of first passage percolation. In École d’été de probabilités de Saint-Flour, XIV—1984125–264. Lecture Notes in Math. 1180. Springer, Berlin, 1986. Zbl0602.60098MR876084
  9. [9] T. Seppäläinen. Large deviations for increasing sequences on the plane. Probab. Theory Related Fields 112 (1998) 221–244. Zbl0919.60040MR1653841

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.