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

### On the partition function of a directed polymer in a random environment

Auteur(s):

Code(s) de Classification MSC:

• 60K99 None of the above but in this section
• 82D30 Random media, disordered materials (including liquid crystals and spin glasses)

Résumé: Consider a $d$-dimensional directed polymer in a Gaussian random environment $(g(i,x))_{i\ge1, x\in \z^d}$, we study its partition function $Z_n(\beta) = \EEE\big( e^{\beta \sum_1^n g(i, S_i)} \big)$ in all dimension $d\ge1$. It is proven that for all $\beta>0$ and $d\ge1$, ${\log Z_n(\beta) \over n}$ converges almost surely to some constant $p(\beta)$ and ${\log Z_n(\beta) \over n}$ satisfies a large deviation principle on $[p(\beta), \infty)$. In the low-dimensional cases ($d=1$ or $d=2$), there is a phase transition for all $\beta>0$: the renormalized partition function $Z_n(\beta) e^{- {\beta^2\over 2} n}$ converges to $0$ and the correlation $\langle \i_{(S^1_n=S^2_n)}\rangle^{(n)}$ of two independent configurations does not converge to $0$.

Mots Clés: Directed polymer in random environment ; partition function ; phase transition ; large deviation principle

Date: 2001-06-08

Prépublication numéro: PMA-665

Postscript file : PMA-665.ps