2013/10/07 by Teodor Banica, Banica, Teodor · 1 citation
Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #graph theory and CDMA systems #math.CO #math.PR
paper · pdf · doi:10.48550/arxiv.1310.1810
Withdrawn by the author - the main findings in this paper are now part of arXiv:1403.2108
openalex publication_date 2013/10/07 · arxiv created 2014/12/02 · arxiv updated 2014/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Associated to a complex Hadamard matrix H∈ MN(\mathbb C) is the complex probability measure μ∈\mathcal P(\mathbb C) describing the distribution of φ(a,b)=<a,Hb>, where a,b∈\mathbb TN are random. This measure is called "glow" of the matrix, due to the analogy with the Gale-Berlekamp switching game, where H,a,b are real. We prove here that: (1) μ becomes complex Gaussian in the N→∞ limit, (2) the universality holds as well at order 2, (3) the order 3 term seems to be quite interesting, particularly for the master Hadamard matrices, (4) in the Fourier matrix case, some of the higher order terms control counting problems for circulant Hadamard matrices.