Communications in Mathematical Physics

, Volume 254, Issue 2, pp 257–287 | Cite as

Brownian Directed Polymers in Random Environment

  • Francis CometsEmail author
  • Nobuo Yoshida


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.


Phase Diagram Partition Function Brownian Motion Lyapunov Exponent Poisson Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Université Paris 7, MathématiquesParisFrance
  2. 2.Division of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan

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