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

Frédérique Watbled (Université Bretagne Sud)

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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-9

Publication Date: November 11, 2012

DOI: 10.1214/ECP.v17-2221

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. MR1006293
  • 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. MR2256483
  • 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. MR1939654
  • Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9 (2003), no. 4, 705--723. MR1996276
  • 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. MR2073332
  • Comets, Francis; Vargas, Vincent. Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267--277. MR2249671
  • Comets, Francis; Yoshida, Nobuo. Brownian directed polymers in random environment. Comm. Math. Phys. 254 (2005), no. 2, 257--287. MR2117626
  • Comets, Francis; Yoshida, Nobuo. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006), no. 5, 1746--1770. MR2271480
  • Giacomin, Giambattista. Random polymer models. Imperial College Press, London, 2007. xvi+242 pp. ISBN: 978-1-86094-786-5; 1-86094-786-7 MR2380992
  • 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 MR1707339
  • 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. MR1908695
  • 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. MR2579463
  • 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 MR2108619
  • 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. MR2568267
  • 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. MR2395475
  • Watbled, Frédérique. Concentration inequalities for disordered models. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 129--140. MR2904479
Bookmark and Share


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