Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60K37 Processes in random environments
- 82D60 Polymers
- 60J65 Brownian motion [See also 58J65]
- 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.)

**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}.

**Mots Clés:** *Directed polymers ; random environment ; fluctuation exponent ; wandering exponent*

**Date:** 2004-02-24

**Prépublication numéro:** *PMA-884*