DOI: 10.1051/jp1:1992278
J. Phys. I France 2 (1992) 2231-2242

Manifolds in random media: two extreme cases

M. Mézard¹ and G. Parisi<sup>2

1</sup>  Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond, 75231 Paris Cedex 05, France
²  Dipartimento di fisica, Università di Roma II, via E. Carnevale, Roma 00173, Italy

(Received 27 May 1992, accepted 7 September 1992)

Abstract
We discuss two special cases of directed manifolds in random media. One is the zero dimensional "toy" model of one particle in a random potential, the other is the limit where the manifold is embedded in a space of large dimensionality. We use the recently introduced Gaussian variational approach in replica space, with replica symmetry breaking, and compare it to known results and simulations of the toy model. In large dimensions ( ) the variational approach is exact. We compute one of the diagrams contributing to the O(1/N) correction.

Résumé
Nous étudions deux cas particuliers de variétés dirigées en milieu aléatoire. L'un est le modèle à zéro dimension d'une particule dans un potentiel aléatoire, l'autre le cas limite où la variété est plongée dans un espace de dimension élevée. Nous utilisons la méthode introduite récemment des variations gaussiennes dans l'espace à des répliques, avec brisure de la symétrie des répliques, et nous la comparons à des résultats connus ainsi qu'à des simulations numériques de ce modèle. En grande dimension ( ) la méthode variationnelle est exacte. Nous calculons un des diagrammes qui contribuent aux corrections en 1/N.

© Les Editions de Physique 1992

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