Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Some new results on Brownian directed polymers in random environment

Auteur(s):

Code(s) de Classification MSC:

• 60K37 Processes in random environments
• 82D60 Polymers
Résumé: We prove some new results on Brownian directed polymers in random environment recently introduced by the authors. The directed polymer in this model is a $d$-dimensional Brownian motion (up to finite time $t$) viewed under a Gibbs measure which is built up with a Poisson random measure on $\R_+ \times \R^d$ (=time $\times$ space). Here, the Poisson random measure plays the role of the random environment which is independent both in time and in space. We prove that (i) For $d \ge 3$ and the inverse temperature $\beta$ smaller than a certain positive value $\beta_0$, the central limit theorem for the directed polymer holds almost surely with respect to the environment. (ii) If $d=1$ and $\beta \neq 0$, the variance of the free energy diverges with a magnitude not smaller than $t^{1/8}$ as $t$ goes to infinity. The argument leading to this result strongly supports the inequalities $\chi(1)\geq 1/5$ for the fluctuation exponent for the free energy, and $\xi(1)\geq 3/5$ for the wandering exponent. We provide necessary background by reviewing some results in the previous paper \cite{CY03}.