Bulk and Blip Spectral Distributions in Combined Random Matrix Ensemble by Saad Waheed ’25

Bulk and Blip Spectral Distributions in Combined Random Matrix Ensembles via Anticommutator Operators by Saad Waheed ’25, Monday December 2, 1:00 – 1:50pm, North Science Building 113, Wachenheim, Mathematics Colloquium

Random Matrix Theory is crucial for analyzing eigenvalue statistics in physics, number theory, and data science. Classical ensembles such as the Gaussian Orthogonal Ensemble exhibit bulk spectral distributions characterized by Wigner’s semicircle law. However, introducing additional symmetries, as exemplified by Palindromic Toeplitz and checkerboard matrices, enriches the spectra with discrete outlier eigenvalues, or “blips,” alongside the bulk. I will be talking about the spectral properties of combined random matrix ensembles formed via the anticommutator operator. Through moment analysis, we develop combinatorial recurrence relations that describe both the bulk and blip distributions in the asymptotic limit. Our results reveal how symmetries govern the interaction between bulk and blip spectral components, which has applications in modeling systems with both continuous and discrete spectral features.