2009/07/29 by Yehonatan Elon, Elon, Yehonatan
Mathematics · Physics and Astronomy · #60G15 #60G60 #Adjacency list #Bounded function #Cardinality (data modeling) #Combinatorics #Computer science #Covariance #Discrete mathematics #Eigenfunction #Eigenvalues and eigenvectors #FOS: Physical sciences #Gaussian #Gaussian process #Geometry and complex manifolds #Invariant (physics) #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics #Tree (set theory) #math-ph #math.MP #msc:60G15 #msc:60G60
paper · pdf · doi:10.48550/arxiv.0907.5065
28 pages, 2 figures. Figure and typos added
openalex publication_date 2009/07/29 · arxiv created 2009/10/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the family of real (generalized) eigenfunctions of the adjacency operator on Td - 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.