Directed polymers on infinite graphs

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.

References in corpus (7)

• Random walks on supercritical percolation clusters
• New bounds for the free energy of directed polymers in dimension 1+1 and 1+2
• Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment
• The Canopy Graph and Level Statistics for Random Operators on Trees
• The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher
• Localization Transition for Polymers in Poissonian Medium
• Carne--Varopoulos bounds for centered random walks

Cited by in corpus (1)

• Absence of weak disorder for directed polymers on supercritical percolation clusters