Electronic Communications in Probability

A note on directed polymers in gaussian environments

Yueyun Hu and Qi-Man Shao

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We study the problem of directed polymers in gaussian environments in $\mathbb{Z}^d$ from the viewpoint of a gaussian family indexed by the set of random walk paths. In the zero-temperature case, we give a numerical bound on the maximum of the Hamiltonian, whereas in the finite temperature case, we establish an equivalence between the "very strong disorder" and the growth rate of the entropy associated to the model

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 50, 518-528.

Accepted: 24 September 2009
First available in Project Euclid: 6 June 2016

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments

Directed polymer gaussian environment

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Hu, Yueyun; Shao, Qi-Man. A note on directed polymers in gaussian environments. Electron. Commun. Probab. 14 (2009), paper no. 50, 518--528. doi:10.1214/ECP.v14-1509. https://projecteuclid.org/euclid.ecp/1465234759

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  • Birkner, Matthias. A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab. 9 (2004), 22–25 (electronic).
  • 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. 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; Yoshida, Nobuo. Directed polymers in random environment are diffusive at weak disorder. Ann. Probab. 34 (2006), no. 5, 1746–1770.
  • Comets, Francis; Vargas, Vincent. Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267–277 (electronic).
  • Gross, Leonard. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061–1083.
  • Cirelʹson, B. S.; Ibragimov, I. A.; Sudakov, V. N. Norms of Gaussian sample functions. Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), pp. 20–41. Lecture Notes in Math., Vol. 550, Springer, Berlin, 1976.
  • Imbrie, J. Z.; Spencer, T. Diffusion of directed polymers in a random environment. J. Statist. Phys. 52 (1988), no. 3-4, 609–626.
  • Johansson, Kurt. Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 (2000), no. 4, 445–456.
  • Lacoin, H. New bounds for the free energy of directed polymers in dimension 1+1 and 1+2. Arxiv arXiv:0901.0699
  • Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120–216, Lecture Notes in Math., 1709, Springer, Berlin, 1999.
  • Ledoux, M. Logarithmic Sobolev inequalities for unbounded spin systems revisited. Séminaire de Probabilités, XXXV, 167–194, Lecture Notes in Math., 1755, Springer, Berlin, 2001.
  • Slepian, David. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 1962 463–501. (24 #A3017)