DIRECTED POLYMERS IN A RANDOM MEDIUM

The problem of a polymer in a random medium can be formulated as follows: consider a regular lattice with on each bond ij a random energy ε{lunate}ij. The energy Ew of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk. Then the problem is to compute the partition function Z = ∑w exp(-Ew/T). Several recent results will be presented: (1) The mean-field version of this problem can be solved exactly and one observes a low-temperature phase similar to the spin-glass phase. (2) For finite-dimensional lattices, one can derive bounds on the transition temperature in high enough dimension. (3) For hierarchical lattices, the problem can be reduced to finding stable laws when one combines random variables in a non-linear way. For example: Z = Z1Z2 + Z3Z4. © 1990.