Dirichlet forms and polymer models based on stable processes

Abstract

In this paper, we are concerned with polymer models based on α-stable processes, where α∈(d2,d∧2) and d stands for dimension. They are attached with a delta potential at the origin and the associated Gibbs measures are parametrized by a constant γ∈R∪{−∞} playing the role of inverse temperature. Phase transition exhibits with critical value γcr=0. Our first object is to formulate the associated Dirichlet form of the canonical Markov process X(γ) induced by the Gibbs measure for a globular state γ>0 or the critical state γ=0. Approach of Dirichlet forms also leads to deeper descriptions of their probabilistic counterparts. Furthermore, we will characterize the behaviour of polymer near the critical point from probabilistic viewpoint by showing that X(γ) is convergent to X(0) as γ↓0 in a certain meaning.