本项目研究关于Cayley图上的均匀格点树、渗流和随机游走的如下问题：（1）Cayley图上均匀格点树的直径及其期望的阶以及相应的尺度极限。（2）顺从Cayley图上的上临界渗流连通分支在其有限时的尾概率衰减行为。（3）群上随机游走的噪声敏感性与Liouville性质之间的关系。（1）的研究能提供对均匀格点树的新认识。（2）的研究能补全Cayley图上的上临界渗流连通分支在其有限时的尾概率衰减行为。（3）的研究能从群上随机游走的Liouville性质的角度刻画相关的噪声敏感性。

In this project, we study the following problems about uniform lattice tree, percolation and random walk on Cayley graphs: (1) The orders of diameter of uniform lattice tree and its expectation, and the scaling limit of the uniform lattice tree on Cayley graphs. (2) Tail probability decay behavior of a connected component in a supercritical percolation on amenable Cayley graphs when the component is finite. (3) Relationship between noise sensitivity of random walks on groups and Liouville property of these random walks. It will provide a new understanding of uniform lattice trees by studying (1), and will complement tail probability decay behavior of a connected component in a supercritical percolation on Cayley graphs when the component is finite by studying (2). And it can characterize noise sensitivity of random walks on groups in term of Liouville property of these random walks by investigating (3).