Manifolds in random media: a variational approach to the spatial probability distribution

We dcvciop a new variaiional scheme to approximate tlie podion dependent spatial probability distribution of a zero dimensional manifold in a random mcdium. This celebrated 't,oy-model' is associated via a. ma,pping with directed polymers in 1+1 dimension, and also describes feat.ures of the commensiirate-incommensurate phase transition. I t consists of a pointlike 'intrrfare' in one dimension subject to a coinbiiiation of a harmonic potential plus a random potential with long range spatial correlations. The variational approach we develop gives far better results for the tail of the spatial distribution than the Harniltonian version, dcvrloped by Mezard and Pirisi, is compared with numerical siniulations for a range of tempcraturcs. This is because the variational parameters arc deteiniined a functions of position. The replica method i s utilized, and solut,ions for the rariationd parmeters are presented. In this paper we limit, ourselves to the rqlica synimetiic solution.