Tree polymers in the infinite volume limit at critical strong disorder

TL;DR

This paper proves the existence of an infinite volume polymer probability at critical strong disorder, where convergence occurs in probability, and provides calculations for the asymptotic variance of the polymer path.

Contribution

It establishes the existence of an infinite volume probability at critical strong disorder and offers calculations supporting a specific formula for the asymptotic variance.

Findings

Existence of infinite volume probability at critical strong disorder.

Convergence in probability rather than almost sure.

Supportive calculations for asymptotic variance formula.

Abstract

The a.s. existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walk. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the a.s. asymptotic variance of the polymer path under strong disorder are also provided.