【作者】 张杰；

【导师】 刘世平；

【作者基本信息】 中国科学技术大学 ， 基础数学， 2022， 硕士

【摘要】 本文首先介绍了 expander,Ollivier-Ricci曲率,基本的遍历定理和熵论。随后介绍了图极限理论和图上的Liouville性质;类比Benjamini-Curien[1,Theorem 3.2]的证明,利用熵论论述了 Liouville性质和熵为零在概率意义下是等价的;说明了非负曲率下Liouville性质是概率意义下几乎处处成立的。最后,补充了Salez[2]文章的细节,特别是将所涉及到的概率论和随机过程的部分,给出了[2,Theorem 2]即非负曲率图中不存在expander的详细的证明过程。更多还原

【Abstract】 We fisrt introduce expander,Ollivier-Ricci curvature,basic ergodic theorems and entropy theory.After that we present the limit theory of graphs and the Liouville property;by analogy with the proof of Benjamini-Curien[1,Theorem 3.2],we show that the Liouville property and zero-entropy are equivalent in the sense of probability by using entropy theory;also,it is discussed that the Liouville property under nonnegative curvature is true almost everywhere in the sense of probability.Last,The details of the Salez[2]are supplemented,especially the parts of probability theory and stochastic process,and the detailed proof of[2,Theorem 2],that is,the graphs of nonnegative curvature have no expander is given.更多还原