Directed polymers on hierarchical lattices with site disorder

We study a polymer model on 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 ββ, and give some accurate results on 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 an upper bound, and a second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent of logZnlogZn, and study the infinite polymer measure in the weak disorder phase.