2024/04/10 by Michail Louvaris, Louvaris, Michail, Daniel T. Wise +3 · 2 citations
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2404.07321
openalex publication_date 2024/04/10 · openalex created_date 2024/04/13 · openalex updated_date 2026/08/03
We prove a concentration result for the leading eigenvalue of the non--backtracking matrix of the configuration model under the assumption of uniformly bounded degrees. Let P denote the limiting degree distribution. Assuming polynomial approximation, we show that as the number of vertices tends to infinity, the leading eigenvalue of the non--backtracking matrix concentrates around (𝔼[P(P-1)])/(𝔼[P]). This quantity corresponds to the mean offspring number of the excess--degree branching process associated with the local limit of the configuration model. As a byproduct of our work we explain how this result can be applied to prove the density of the growth rates of the subgroups of the free group.