Minimal supporting subtrees for the free energy of polymers on disordered trees

Details Minimal supporting subtrees for the free energy of polymers on disordered trees Minimal supporting subtrees for the free energy of polymers on disordered trees

2008-06-20

arxiv.org/abs/0806.3430v1

Mathematics

Probability

22 pages, 3 figures, to be published in the special issue of JMP, 'Statistical Mechanics on Random Structures'

Scientific paper

10.1063/1.2962981

We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model undergoes a phase transition with respect to its localization behaviour. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree is decreasing to zero as the temperature decreases to the critical value. At the critical value and all lower temperatures, a single polymer suffices to support the free energy. Our proofs rely on elegant martingale methods adapted from the theory of branching random walks.

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