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In this paper, we are concerned with polymer models based on $\alpha$-stableprocesses, where $\alpha\in (\frac{d}{2},d\wedge 2)$ and $d$ stands fordimension. They are attached with a delta potential at the origin and theassociated Gibbs measures are parametrized by a constant $\gamma$ playing therole of inverse temperature. Phase transition exhibits with critical value$\gamma_{cr}=0$. Our first object is to formulate the associated Dirichlet formof the canonical Markov process $X^{(\gamma)}$ induced by the Gibbs measure fora globular state $\gamma>0$ or the critical state $\gamma=0$. Approach ofDirichlet forms also leads to deeper descriptions of probabilistic counterpartsof globular and critical states. Furthermore, we will characterize thebehaviour of polymer near the critical point from probabilistic viewpoint byshowing that $X^{(\gamma)}$ is convergent to $X^{(0)}$ as $\gamma\downarrow 0$in a certain meaning.

Dirichlet forms and polymer models based on stable processes