Gaussian Waves on the Regular Tree

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.