正则树上的高斯波
Gaussian Waves on the Regular Tree
摘要：
我们考虑$T_d$ -$d$ -正则树上邻接算子的实（广义）本征函数族.我们证明了一个唯一的不变高斯过程的存在性的合奏，并明确推导出其协方差算子。我们调查的过程的水平集的典型结构。特别是，我们证明了熵排斥的水平集是一致有界的，并证明了存在一个临界阈值，在此阈值以上的水平集都是有限的基数和低于它的无限组件几乎肯定会出现。
英文摘要：
We consider the family of real (generalized) eigenfunctions of the adjacency operator on $T_d$ - the $d$-regular tree. We show the existence of a unique invariant Gaussian process on the ensemble and derive explicitly its covariance operator. We investigate the typical structure of level sets of the process. In particular we show that the entropic repulsion of the level sets is uniformly bounded and prove the existence of a critical threshold, above which the level sets are all of finite cardinality and below it an infinite component appears almost surely.
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