, Volume 254, Issue 2, pp 257-287
Date: 14 Oct 2004

Brownian Directed Polymers in Random Environment

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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.

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.