Electronic Communications in Probability

Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment

Frédérique Watbled

Full-text: Open access

Abstract

We give sharp estimate for the free energy of directed polymers in random environment in dimension 1+1. This estimate was known for a Gaussian environment, we extend it to the case where the law of the environment is infinitely divisible.

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 53, 9 pp.

Dates
Accepted: 11 November 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263186

Digital Object Identifier
doi:10.1214/ECP.v17-2221

Mathematical Reviews number (MathSciNet)
MR2999981

Zentralblatt MATH identifier
1306.60154

Subjects
Primary: 60K37: Processes in random environments
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 60E07: Infinitely divisible distributions; stable distributions

Keywords
directed polymers in random environment free energy infinite divisibility FKG inequality

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Watbled, Frédérique. Sharp asymptotics for the free energy of 1+1 dimensional directed polymers in an infinitely divisible environment. Electron. Commun. Probab. 17 (2012), paper no. 53, 9 pp. doi:10.1214/ECP.v17-2221. https://projecteuclid.org/euclid.ecp/1465263186


Export citation

References

  • T. Alberts, K. Khanin, and J. Quastel, Intermediate Disorder Regime for 1+1 Dimensional Directed Polymers, arXiv:1202.4398 (2012).
  • Bolthausen, Erwin. A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 (1989), no. 4, 529–534.
  • Carmona, Philippe; Guerra, Francesco; Hu, Yueyun; Menjane, Olivier. Strong disorder for a certain class of directed polymers in a random environment. J. Theoret. Probab. 19 (2006), no. 1, 134–151.
  • Carmona, Philippe; Hu, Yueyun. On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 (2002), no. 3, 431–457.
  • Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9 (2003), no. 4, 705–723.
  • Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Probabilistic analysis of directed polymers in a random environment: a review. Stochastic analysis on large scale interacting systems, 115–142, Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, 2004.
  • Comets, Francis; Vargas, Vincent. Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267–277.
  • Comets, Francis; Yoshida, Nobuo. Brownian directed polymers in random environment. Comm. Math. Phys. 254 (2005), no. 2, 257–287.
  • Comets, Francis; Yoshida, Nobuo. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006), no. 5, 1746–1770.
  • Giacomin, Giambattista. Random polymer models. Imperial College Press, London, 2007. xvi+242 pp. ISBN: 978-1-86094-786-5; 1-86094-786-7
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6
  • Guerra, Francesco; Toninelli, Fabio Lucio. Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model. J. Math. Phys. 43 (2002), no. 7, 3704–3716.
  • Lacoin, Hubert. New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$. Comm. Math. Phys. 294 (2010), no. 2, 471–503.
  • Liggett, Thomas M. Interacting particle systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005. xvi+496 pp. ISBN: 3-540-22617-6
  • Liu, Quansheng; Watbled, Frédérique. Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment. Stochastic Process. Appl. 119 (2009), no. 10, 3101–3132.
  • T. Sasamoto and H. Spohn, The 1 + 1-dimensional Kardar-Parisi-Zhang equation and its universality class, Journal of Statistical Mechanics: Theory and Experiment 11 (2010), 13.
  • Toninelli, Fabio Lucio. A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 (2008), no. 2, 389–401.
  • Watbled, Frédérique. Concentration inequalities for disordered models. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 129–140.