Directed polymers on infinite graphs

TL;DR

This paper investigates the behavior of directed polymers on various infinite graphs, establishing conditions for different disorder phases and analyzing specific cases like random walks on trees, revealing surprising counterexamples.

Contribution

It extends the directed polymer model to general graphs, providing criteria for disorder phases and exploring diverse graph structures including trees and counterexamples.

Findings

Conditions for weak and strong disorder phases established.

Analysis of random walks on trees like Galton-Watson trees.

Counterexamples to intuitive extensions of lattice results.

Abstract

We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder, in terms of properties of the graph and of the random walk. We study in some detail (biased) random walk on various trees including the Galton Watson trees, and provide a range of other examples that illustrate counter-examples to intuitive extensions of the $\mathbb Z^d$/SRW result.