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ABSTRACT: 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(a) reminiscent of that used in dynamical
systems. We compare our results with numerical simulations. |