Disordered systems and Burger's turbulence

Abstract
In order to celebrate the memory of my friend Giovanni, I could not imagine a better topic than one which lies at the interface between the studies of turbulence and tlie statistical mechanics of disordered systems, his two favorite fields in physics. I have recently written a paper on this subject, in the Proceedings of the International Conference of Mathematical Physics 1997, and what I have written here is basically the content of this recent paper. The basic properties of scaling and intermittency in fully developped turbulence can be studied in a lot of details in the case of stirred Burgers turbulence, using the mapping between Burgers' equation and the problem of a directed polymer in a random medium. In the limit of large dimenisions, a replica symmetry breaking solution of the polymer problem provides the full probability distribution of the velocity difference u(r) between points separated by a distance r much smaller than the correlation length of the forcing. This exhibits a very strong intermittency which is related to regions of shock waves, in the fluid, and to the existence of metastable states in the directed polymer problem.