2005/04/30 by Michael Aizenman, Robert Sims, Simone Warzel · 101 citations
Mathematics · Physics and Astronomy · #Absolute continuity #Bounded function #Combinatorics #Discrete mathematics #Graph #Independent and identically distributed random variables #Laplace operator #Limit (mathematics) #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum #Quantum chaos and dynamical systems #Quantum graph #Quantum mechanics #Random graph #Random variable #Spectral Theory in Mathematical Physics #Tree (set theory) #advanced mathematical theories #cond-mat.dis-nn #math-ph #math.MP #math.SP
paper · pdf · doi:10.1007/s00220-005-1468-5
published in Communications in Mathematical Physics 264(2), 371-389 (Springer Science+Business Media)
arxiv created 2005/06/05 · openalex publication_date 2005/11/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the Laplacian on a rooted metric tree graph with branching number K ≥ 2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.