2003/01/23 by O. Golinelli, Golinelli, O. · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0301437
13 pages, 9 fig
arxiv created 2003/01/23 · openalex publication_date 2003/01/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an analysis of the spectral density of the adjacency matrix of large random trees. We show that there is an infinity of delta peaks at all real numbers which are eigenvalues of finite trees. By exact enumerations and Monte-Carlo simulations, we have numerical estimations of the heights of peaks. In the large tree limit, the sum of their heights is 0.19173 +- 0.00005. Moreover all associated eigenvectors are strictly localized on a finite number of nodes. The rest of the spectral density is a function which vanishes at all positions of peaks, which are a dense subset of real numbers: so this function is almost everywhere discontinuous. Keywords: random tree, spectral density, density of states, adjacency matrix, localization, delta peak.