A Local limit theorem for directed polymers in random media: the continuous and the discrete case
نویسندگان
چکیده
In this article, we consider two models of directed polymers in random environment: a discrete model in a general random environment and a continuous model. We consider these models in dimension greater or equal to 3 and we suppose that the normalized partition function is bounded in L (the ”high” temperature case). Under these assumptions, Sinai proved in [11] a local limit theorem for the discrete model, using a perturbation expansion. In this article, we give a new method for proving Sinai’s local limit theorem. This new method can be transposed to the continuous setting in which we prove a similar local limit theorem. Resumé: Dans cet article, on considère deux modèles de polymères dirigés en environnement aléatoire: un modèle discret en environnement aléatoire général et un modèle continu. On considère ces modèles en dimension supérieur ou égale à 3 et on suppose que la fonction de partition renormalisée est bornée dans L (cela correspond au cas de ”haute” température). Sous ces hypothèses, Sinai a montré dans [11] un théorème limite locale pour le modèle discret en utilisant un développement en perturbation. Dans cet article, on donne une nouvelle méthode pour démontrer le théorème limite locale ci-dessus. Cette nouvelle méthode peut être transposée au cas continu dans lequel on montre un théorème limite locale similaire. MSC: 60K37;60F05;82B44;82D60
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