In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of the space of harmonic functions (SHF) for a transfer operator. This is accomplished by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example of a potential having a phase transition where the Perron-Frobenius eigenvector space has dimension two. We discuss entropy and equilibrium states, in this general setting, and show that if the SHF is non-trivial, then the associated equilibrium states have full support. We also obtain a weak invariance principle in cases where the spectral gap property is absent. As a consequence, a functional central limit theorem for non-local observables of the Dyson model is obtained.

在本文中，我们展示了统计力学中的相变与转移算子的谐波函数空间（SHF）的维数之间的新关系。这是通过将经典的Ruelle-Perron-Frobenius理论扩展到在有限或无限（不可数）字母上定义的低规则势域实现的。我们还给出了一个具有相变的电势示例，其中Perron-Frobenius特征向量空间的维数为2。我们讨论了在这种一般情况下的熵和平衡状态，并表明如果SHF不平凡，则相关的平衡状态将得到充分支持。在不存在谱隙特性的情况下，我们还获得了弱不变性原理。结果，获得了戴森模型的非局部可观量的函数中心极限定理。