The multifractal nature of the probability distribution for the head of a directed polymer in a (1+1)-dimensional disordered medium is derived analytically. This is achieved by using a mapping of the model into a corresponding ‘‘toy’’ model which consists of a classical particle in a combination of a harmonic potential and a long-ranged random potential. We use the solution of the latter problem that incorporates replica-symmetry breaking within the framework of a variational approximation. The results are expressed in terms of a distribution f(α) reminiscent of that used in dynamical systems. We compare our results with numerical simulations.

• Received 10 March 1993

DOI:https://doi.org/10.1103/PhysRevE.48.161