Abstract
We study the free energy of the directed polymer in a random environment model in dimension 1 + 1 and 1 + 2. For dimension one, we improve the statement of Comets and Vargas in [8] concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension two, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albeverio S., Zhou X.: A martingale approach to directed polymers in a random environment. J. Theor. Probab. 9, 171–189 (1996)
Bertin, P.: Free energy for Linear Stochastic Evolutions in dimension two. Preprint 2009
Bolthausen E.: A note on diffusion of directed polymer in a random environment. Commun. Math. Phys. 123, 529–534 (1989)
Carmona P., Hu Y.: On the partition function of a directed polymer in a random Gaussian environment. Probab. Theor. Relat. Fields 124(3), 431–457 (2002)
Carmona P., Hu Y.: Strong disorder implies strong localization for directed polymers in a random environment. ALEA 2, 217–229 (2006)
Comets F., Shiga T., Yoshida N.: Directed Polymers in a random environment: strong disorder and path localization. Bernouilli 9(4), 705–723 (2003)
Comets F., Shiga T., Yoshida N.: Probabilistic analysis of directed polymers in a random environment: a review. Adv. Stud. Pure Math. 39, 115–142 (2004)
Comets F., Vargas V.: Majorizing multiplicative cascades for directed polymers in random media. ALEA 2, 267–277 (2006)
Comets F., Yoshida N.: Directed polymers in a random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006)
Derrida B., Giacomin G., Lacoin H., Toninelli F.L.: Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys. 287, 867–887 (2009)
Feller W.: An Introduction to Probability Theory and Its Applications Volume II. John Wiley & Sons, Inc, New York (1966)
Gantert N., Zeitouni O.: Large and moderate deviations for local time of a reccurent Markov chain on Z 2. Ann. Inst. H. Poincaré Probab. Statist. 34, 687–704 (1998)
Giacomin, G.: Random Polymer Models. IC press, London: World Scientific, 2007
Giacomin, G., Lacoin, H., Toninelli, F.L.: Hierarchical pinning models, quadratic maps and quenched disorder. To appear in Probab. Theory. Rel. Fields, doi:10.1007/s00440-009-0205-y
Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. to appear in Commun. Pure Appl. Math., doi:10.1002/cpa.20301
Giacomin, G., Lacoin, H., Toninelli, F.L.: Disorder relevance at marginality and critical point shift. http://arxiv.org/abs/0906.1942v1[math-ph], 2009
Grimmett, G.: Percolation. Second Edition, Grundlehren der Mathematischen Wissenschaften 321, Berlin: Springer-Verlag, 1999
Huse D.A., Henley C.L.: Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54, 2708–2711 (1985)
Imbrie J.Z., Spencer T.: Diffusion of directed polymer in a random environment. J. Stat. Phys. 52(3/4), 608–626 (1988)
Kaigh W.D.: An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4, 115–121 (1976)
Lacoin, H., Moreno, G.: Directed polymer on hierarchical lattice with site disorder. http://arxiv.org/abs/0906.0992v1[math.PR], 2009
Liu Q., Watbled F.: Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment. Stoch. Proc. Appl. 119(10), 3101–3132 (2009)
Song R., Zhou XY.: A remark on diffusion of directed polymers in random environment. J. Stat. Phys. 85(1/2), 277–289 (1996)
Toninelli F.L.: A replica coupling-approach to disordered pining models. Commun. Math. Phys. 280, 389–401 (2008)
Toninelli F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. To appear in Electron. Journal Probab. 14, 531–547 (2009)
Vargas V.: Strong localization and macroscopic atoms for directed polymer. Probab. Theor. Relat. Fields 134(3/4), 391–410 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. L. Toninelli
Rights and permissions
About this article
Cite this article
Lacoin, H. New Bounds for the Free Energy of Directed Polymers in Dimension 1 + 1 and 1 + 2. Commun. Math. Phys. 294, 471–503 (2010). https://doi.org/10.1007/s00220-009-0957-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0957-3