T-Martingales, size biasing, and tree polymer cascades

Abstract

Polymer models generally refer to random paths having probabilities induced by a random potential. In the case of tree polymers, the paths are defined by connecting vertices to a root of a binary tree, with probabilities given by a random multiplicative cascade normalized to a probability. The basic theory then concerns almost sure probability laws governing (asymptotically) long polymer paths. Weak and strong disorder refer to events in which the (non-normalized) cascade lives or dies, respectively. An almost sure central limit theorem (clt) is established in the full range of weak disorder, extending early results of Bolthausen. Also, almost sure Laplace large deviation rates are obtained under both disorder types. Open problems are included along the way.