Gaussian Waves on the Regular Tree

Abstract

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.

28 pages, 2 figures. Figure and typos added

References in corpus (6)

• Nodal domains statistics - a criterion for quantum chaos
• Percolation model for nodal domains of chaotic wave functions
• The Canopy Graph and Level Statistics for Random Operators on Trees
• Random wave functions and percolation
• Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
• Geometric characterization of nodal domains: the area-to-perimeter ratio

Cited by in corpus (7)

• Local Kesten--McKay law for random regular graphs
• Edge rigidity and universality of random regular graphs of intermediate degree
• Cycles and eigenvalues of sequentially growing random regular graphs
• Spectrum of Random $d$-regular Graphs Up to the Edge
• Many nodal domains in random regular graphs
• The threshold for SDP-refutation of random regular NAE-3SAT
• Level sets percolation on chaotic graphs