2003/12/10 by Christopher Hammond, Steven J. Miller · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Central limit theorem #Circular law #Combinatorics #Convergence of random variables #Diophantine equation #Distribution (mathematics) #Eigenvalues and eigenvectors #Gaussian #Mathematical analysis #Mathematics #Matrix Theory and Algorithms #Measure (data warehouse) #Probability measure #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Random variable #Statistics #Toeplitz matrix #math.PR #math.ST #msc:15A52 #msc:60F99 #msc:62H10 #stat.TH
paper · pdf · doi:10.1007/s10959-005-3518-5
published as Journal of Theoretical Probability, Vol. 18 (2005), no. 3, 537 - 566 · 24 pages, 3 figures
arxiv created 2003/12/10 · openalex publication_date 2005/07/01 · arxiv updated 2010/11/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of p. This distribution's moments are almost those of the Gaussian's; the deficit may be interpreted in terms of Diophantine obstructions. With a little more work, we obtain almost sure convergence. An investigation of spacings between adjacent normalized eigenvalues looks Poissonian, and not GOE.