Directed polymers on hierarchical lattices with site disorder

Abstract

We study a polymer model oil hierarchical lattices very close to the one introduced and studied in Derrida and Griffith (1989) [19] and Cook and Derrida (1989) [16]. For this model, we prove the existence of free energy and derive the necessary and Sufficient condition for which very strong disorder holds for all beta, and give some accurate results oil the behavior of the free energy at high temperature. We obtain these results by using a combination of fractional moment method and change of Measure over the environment to obtain all upper bound, and a second moment method to get a lower bound. We also get lower bounds oil the fluctuation exponent of log Z(n), and Study the infinite polymer measure in the weak disorder phase. (C) 2009 Elsevier B.V. All rights reserved.