2020/08/13 by Nalini Anantharaman, Anantharaman, Nalini, Maxime Ingremeau +5
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2008.05709
openalex publication_date 2020/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notion of Benjamini-Schramm convergence for quantum graphs.\nThis notion of convergence, intended to play the role of the already existing\nnotion for discrete graphs, means that the restriction of the quantum graph to\na randomly chosen ball has a limiting distribution. We prove that any sequence\nof quantum graphs with uniformly bounded data has a convergent subsequence in\nthis sense. We then consider the empirical spectral measure of a convergent\nsequence (with general boundary conditions and edge potentials) and show that\nit converges to the expected spectral measure of the limiting random rooted\nquantum graph. These results are similar to the discrete case, but the proofs\nare significantly different.\n