Directed Polymers on Infinite Graphs

We study the directed polymer model for general graphs (beyond 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 Z(d)/SRW result.

Automated summary

This study examines the directed polymer model for general graphs and random walks beyond Z(d), providing conditions for the presence or absence of weak disorder phases, L-2 regions, and very strong disorder based on graph and random walk properties. The study delves into (biased) random walks on various trees, including Galton-Watson trees, and offers a range of examples that demonstrate counterexamples to intuitive extensions of the Z(d)/SRW results.