# Dirichlet forms and polymer models based on stable processes

## TL;DR

This paper investigates polymer models based on stable processes, formulates their Dirichlet forms, and analyzes the behavior of associated Markov processes near the critical point, revealing phase transition properties.

## Contribution

It introduces a Dirichlet form framework for polymer models with stable processes and characterizes the convergence of the Markov processes at the critical point.

## Key findings

- Dirichlet form formulation for polymer models
- Characterization of globular and critical states
- Convergence of Markov processes as parameter approaches critical value

## Abstract

In this paper, we are concerned with polymer models based on $\alpha$-stable processes, where $\alpha\in (\frac{d}{2},d\wedge 2)$ 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 $\gamma$ playing the role of inverse temperature. Phase transition exhibits with critical value $\gamma_{cr}=0$. Our first object is to formulate the associated Dirichlet form of the canonical Markov process $X^{(\gamma)}$ induced by the Gibbs measure for a globular state $\gamma>0$ or the critical state $\gamma=0$. Approach of Dirichlet forms also leads to deeper descriptions of probabilistic counterparts of globular and critical states. Furthermore, we will characterize the behaviour of polymer near the critical point from probabilistic viewpoint by showing that $X^{(\gamma)}$ is convergent to $X^{(0)}$ as $\gamma\downarrow 0$ in a certain meaning.