Directed Polymers in Complex-Valued Random Environments on the Tree

TL;DR

This paper extends the analysis of directed polymers in complex-valued random environments on trees by removing independence assumptions and strengthening convergence results, revealing a richer phase diagram with a third regime.

Contribution

It generalizes previous models by relaxing independence assumptions and proves almost sure convergence of free energy, simplifying existing proofs.

Findings

Identification of a third phase regime due to complex phases

Almost sure convergence of free energy under mild conditions

Simplification of previous analytical arguments

Abstract

We consider a model of directed polymers on regular trees with complex-valued random weights introduced by Cook and Derrida [CD90] and studied mathematically by Derrida, Evans and Speer [DES93]. In addition to the usual weak-disorder and strong-disorder regimes, the phase diagram of the model contains a third region due to the effects of the random phases that cannot be observed in the model with positive weights. In this work, we extend the results of [DES93] in two directions. First, we remove the hypothesis on the independence of the random radii and random phases of the environment from most of the phase diagram, and second, under mild assumptions on the law of the environment, we strengthen the convergence of the free energy (shown to hold in probability in [DES93]) to an almost sure convergence. Along the way, many of the arguments of [DES93] are simplified.