Directed Polymers on Disordered Hierarchical Lattices

We study the distribution of ground-state energies of directed polymers on disordered hierarchical lattices. The problem can be reduced to finding the stable distribution when one combines random variables in a nonlinear way (for example, e = min ( e 1 + e 2 , e 3 + e 4 )). The ground-state energy fluctuations of a polymer of length L increase like L ω . We calculate ω numerically for a whole family of hierarchical lattices. In a limiting case, we present a perturbation theory for the ground-state energy and the exponent ω, which agrees well with the numerical results.