 0437 Francis COMETS, Nobuo YOSHIDA
 Some new results on
Brownian Directed Polymers in Random Environment
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Feb 13, 04

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Abstract. 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 fluctuation 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:
``Brownian Directed Polymers in Random Environment''
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