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On the dimension of the space of harmonic functions on transitive shift spaces
Advances in Mathematics ( IF 1.5 ) Pub Date : 2021-04-21 , DOI: 10.1016/j.aim.2021.107758
L. Cioletti , L. Melo , R. Ruviaro , E.A. Silva

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.



中文翻译:

关于传递移位空间上调和函数空间的维数

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。

更新日期:2021-04-21
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