Abstract
We study the thermodynamics of a continuous model of directed polymers in random environment. The environment is given by a space-time Poisson point process, whereas the polymer is defined in terms of the Brownian motion. We mainly discuss: (i) The normalized partition function, its positivity in the limit which characterizes the phase diagram of the model. (ii) The existence of quenched Lyapunov exponent, its positivity, and its agreement with the annealed Lyapunov exponent; (iii) The longitudinal fluctuation of the free energy, some of its relations with the overlap between replicas and with the transversal fluctuation of the path.
The model considered here, enables us to use stochastic calculus, with respect to both Brownian motion and Poisson process, leading to handy formulas for fluctuations analysis and qualitative properties of the phase diagram. We also relate our model to some formulation of the Kardar-Parisi-Zhang equation, more precisely, the stochastic heat equation. Our fluctuation results are interpreted as bounds on various exponents and provide a circumstantial evidence of super-diffusivity in dimension one. We also obtain an almost sure large deviation principle for the polymer measure.
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Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)
Birman, M.Š., Solomjak, M. Z.: Piecewise polynomial approximations of functions of classes W p α. (Russian) Mat. Sb. (N.S.) 73(115), 331–355 (1967) English translation: Math. USSR-Sb. 2, 295–317 (1967)
Bolthausen, E.: A note on diffusion of directed polymers in a random environment. Commun. Math. Phys. 123, 529–534 (1989)
Borodin, A.N., Salminen, P.: Handbook of Brownian Motion–Facts and Formulae. 2nd Ed., Basel-Boston-Berlin: Birkhäuser Verlag 2002
Carmona, P., Hu Y.: On the partition function of a directed polymer in a random environment. Probab. Theory Related Fields 124, 431–457 (2002)
Comets, F.: The martingale method for mean-field disordered systems at high temperature. In: Mathematical aspects of spin glasses and neural networks, Progr. Probab. 41, Birkhäuser, Boston: 1998 pp. 91–113,
Comets, F., Shiga, T., Yoshida, N.: Directed Polymers in Random Environment: Path Localization and Strong Disorder. Bernoulli 9, 705–723 (2003)
Dembo, A., Zeitouni, O.: Large Deviation Techniques and Applications. 2nd Ed. Berlin-Heidelberg- New York: Springer Verlag, 1998.
Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51, 817–840 (1988)
Durrett, R.: Probability-Theory and Examples. 2nd Ed., Pacific Grove, CA: Duxbury Press, 1995
Fisher, D.S., Huse, D.A.: Directed paths in random potential. Phys. Rev. B 43, 10,728–10,742 (1991)
Goodman, V., Kuelbs, J.: Rates of clustering in Strassen’s LIL for Brownian motion. J. Theoret. Probab. 4, 285–309 (1991)
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)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes (2nd ed.), Amsterdam / Kodansha,Tokyo: North-Holland, 1989
Imbrie, J.Z., Spencer, T.: Diffusion of directed polymer in a random environment. J. Stat. Phys. 52(3–4), 609–626 (1998)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamical scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Kesten, H.: Aspect of first passage percolation. In: École d’Éte de Probabilités de Saint-Flour XIV, Springer Lecture Notes in Mathematics 1180, Berlin-Heidelberg-New York, 1986, pp. 126–263
Krug, H., Spohn, H.: Kinetic roughening of growing surfaces. In: Solids Far from Equilibrium, C. Godrèche, ed., Cambridge: Cambridge University Press, 1991
Licea, C., Newman, C., Piza, M.: Superdiffusivity in first-passage percolation. Probab. Theory Related Fields 106(4), 559–591 (1996)
Mejane, O.: Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 40, 299–308 (2004)
Newman, C., Piza, M.: Divergence of shape fluctuations in two dimensions. Ann. Probab. 23(3), 977–1005 (1995)
Petermann, M.: Superdiffusivity of directed polymers in random environment. Ph.D. Thesis Univ. Zürich (2000)
Piza, M.S.T.: Directed polymers in a random environment: some results on fluctuations. J. Statist. Phys. 89(3-4), 581–603 (1997)
Rockafeller, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1970
Song, R., Zhou, X.Y.: A remark on diffusion on directed polymers in random environment. J. Statist. Phys. 85(1-2), 277–289 (1996)
Stoyan, D. Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. New York: John Wiley & Sons, 1987
Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics, Berlin-Heidelberg-New York: Springer, 1998
Watson, G.N.: A Treatise on the Theory of Bessel Functions. 2nd ed., Cambridge: Cambridge University Press, 1958
Wu, Liming.: A new modified logarthmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118, 428–438 (2000)
Wüthrich, M.V.: Scaling identity for crossing Brownian motion in a Poissonian potential. Probab. Theory Related Fields 112(3), 299–319 (1998)
Wüthrich, M.V.: Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Probab. 26(3), 1000–1015 (1998)
Wüthrich, M.V.: Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. H. Poincaré Probab. Statist. 34(3), 279–308 (1998)
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Communicated by H. Spohn
Partially supported by CNRS (UMR 7599 Probabilités et Modèles Aléatoires)
Partially supported by JSPS Grant-in-Aid for Scientific Research, Wakatekenkyuu (B) 14740071
Acknowledgement The authors would like to thank Tokuzo Shiga for his careful reading of an earlier version of the manuscript, and two anonymous referees for indicating some obscure points in the manuscript, for suggesting improvements and references.
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Comets, F., Yoshida, N. Brownian Directed Polymers in Random Environment. Commun. Math. Phys. 254, 257–287 (2005). https://doi.org/10.1007/s00220-004-1203-7
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DOI: https://doi.org/10.1007/s00220-004-1203-7