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

TL;DR

This paper investigates the minimal supporting subtrees for the free energy of directed polymers on disordered trees, revealing phase transition behavior and how the supporting structure changes with temperature.

Contribution

It introduces a detailed analysis of the minimal supporting subtrees in polymer models on trees, highlighting the phase transition and the behavior at critical temperature using martingale methods.

Findings

Supportive subtree growth rate decreases to zero as temperature approaches critical

At critical and lower temperatures, a single polymer supports the free energy

High-temperature phase involves a subtree with positive exponential growth rate

Abstract

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.