Polymers on disordered hierarchical lattices: A nonlinear combination of random variables

The problem of directed polymers on disordered hierarchical and hypercubic lattices is considered. For the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way. We present the results of numerical simulations of two hierarchical lattices, finding evidence of a phase transition in one case. For a limiting case we extend the perturbation theory developed by Derrida and Griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties (overlaps) of the model. In this limit we obtain an interpolation formula, allowing one to obtain the noninteger moments of the partition function from the integer moments. We obtain bounds for the transition temperature for hierarchical and hypercubic lattices, and some similarities between the problem on the two different types of lattice are discussed.