New bounds for the free energy of directed polymers in dimension \(1+1\) and \(1+2\). (English) Zbl 1227.82098

The paper deals with the directed polymer model introduced by D. A. Huse and C. L. Henley [Phys. Rev. Letter. 54, No. 25, 2708–2711 (1985), doi:10.1103/PhysRevLett.54.2708] (in dimension \(1+1\)) and it intents to investigate impurity-induced domain-wall roughening in the 2D-Ising model. The author improves previous results on strong disorder in the dimension one and two respectively, and solves a conjecture on the existence of strong disorder at all temperatures.


82D60 Statistical mechanics of polymers
60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] Albeverio S., Zhou X.: A martingale approach to directed polymers in a random environment. J. Theor. Probab. 9, 171–189 (1996) · Zbl 0837.60069
[2] Bertin, P.: Free energy for Linear Stochastic Evolutions in dimension two. Preprint 2009
[3] Bolthausen E.: A note on diffusion of directed polymer in a random environment. Commun. Math. Phys. 123, 529–534 (1989) · Zbl 0684.60013
[4] 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) · Zbl 1015.60100
[5] Carmona P., Hu Y.: Strong disorder implies strong localization for directed polymers in a random environment. ALEA 2, 217–229 (2006) · Zbl 1111.60069
[6] Comets F., Shiga T., Yoshida N.: Directed Polymers in a random environment: strong disorder and path localization. Bernouilli 9(4), 705–723 (2003) · Zbl 1042.60069
[7] 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) · Zbl 1114.82017
[8] Comets F., Vargas V.: Majorizing multiplicative cascades for directed polymers in random media. ALEA 2, 267–277 (2006) · Zbl 1105.60074
[9] Comets F., Yoshida N.: Directed polymers in a random environment are diffusive at weak disorder. Ann. Probab. 34(5), 1746–1770 (2006) · Zbl 1104.60061
[10] 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) · Zbl 1226.82028
[11] Feller W.: An Introduction to Probability Theory and Its Applications Volume II. John Wiley & Sons, Inc, New York (1966) · Zbl 0138.10207
[12] 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) · Zbl 0910.60013
[13] Giacomin, G.: Random Polymer Models. IC press, London: World Scientific, 2007 · Zbl 1125.82001
[14] 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 · Zbl 1190.60093
[15] 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 · Zbl 1189.60173
[16] Giacomin, G., Lacoin, H., Toninelli, F.L.: Disorder relevance at marginality and critical point shift. http://arxiv.org/abs/0906.1942v1[math-ph] , 2009 · Zbl 1210.82036
[17] Grimmett, G.: Percolation. Second Edition, Grundlehren der Mathematischen Wissenschaften 321, Berlin: Springer-Verlag, 1999
[18] 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)
[19] Imbrie J.Z., Spencer T.: Diffusion of directed polymer in a random environment. J. Stat. Phys. 52(3/4), 608–626 (1988) · Zbl 1084.82595
[20] Kaigh W.D.: An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4, 115–121 (1976) · Zbl 0332.60047
[21] Lacoin, H., Moreno, G.: Directed polymer on hierarchical lattice with site disorder. http://arxiv.org/abs/0906.0992v1[math.PR] , 2009 · Zbl 1193.60118
[22] 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) · Zbl 1177.60043
[23] Song R., Zhou XY.: A remark on diffusion of directed polymers in random environment. J. Stat. Phys. 85(1/2), 277–289 (1996) · Zbl 0924.60053
[24] Toninelli F.L.: A replica coupling-approach to disordered pining models. Commun. Math. Phys. 280, 389–401 (2008) · Zbl 1207.82026
[25] Toninelli F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. To appear in Electron. Journal Probab. 14, 531–547 (2009) · Zbl 1189.60186
[26] Vargas V.: Strong localization and macroscopic atoms for directed polymer. Probab. Theor. Relat. Fields 134(3/4), 391–410 (2008) · Zbl 1113.60097
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