  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.
