Directed Polymers on Hierarchical Lattices with site disorder.

Hubert Lacoin (1) , Gregorio Moreno Flores (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We study a polymer model on hierarchical lattices very close to the one introduced and studied in papers by Derrida and Griffith, and by Cook and Derrida. 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 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 second moment method to get a lower bound. We also get lower bounds on the fluctuation exponent of $\log Z_n$, and study the infinite polymer measure in the weak disorder phase.

Keywords: Hierarchical Models; Free Energy; Dynamical System; Directed Polymers

Domains: Probability [math.PR]

Hubert Lacoin, Gregorio Moreno Flores. Directed Polymers on Hierarchical Lattices with site disorder.. 2009. ⟨hal-00391412⟩