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.

中文翻译：

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基于稳定过程的狄利克雷形式和聚合物模型

在本文中，我们关注基于 $\alpha$ 稳定过程的聚合物模型，其中 $\alpha\in (\frac{d}{2},d\wedge 2)$ 和 $d$ 代表维度。它们在原点处附有 delta 电位，并且相关的 Gibbs 测度由一个常数 $\gamma$ 参数化，该常数扮演逆温度的角色。相变具有临界值 $\gamma_{cr}=0$。我们的第一个目标是为球状状态 $\gamma>0$ 或临界状态 $\gamma=0$ 制定由 Gibbs 度量引起的规范马尔可夫过程 $X^{(\gamma)}$ 的关联狄利克雷形式. Dirichlet 形式的方法也导致对球状和临界状态的概率对应物的更深入描述。此外，