Inhomogeneous random matrices have identical universal edge statistics if their variance-profile Markov chains satisfy short-to-long comparability, enabling analysis of band matrices, orbital models, and Hankel profiles in subcritical and critical regimes.
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Spielman's Laplacian eigenratio conjecture is disproved for infinitely many d>2 using Ramanujan graphs but verified for d≤2 and regular graphs, with stronger results and resolutions of two other conjectures.
Proves that SO(3) lattice Yang-Mills theory fails Wilson's confinement criterion at strong coupling.
Proves eigenvalue bounds implying that isogeny graphs of supersingular elliptic curves are Ramanujan and studies their spectral distribution, components, automorphisms, and links to modular forms.
Develops a mixed quantization technique for graph vector bundles and applies it to the Alon-Boppana bound, Kesten-McKay law, asymptotic determinant, quantum ergodicity, zero divisor convergence, and Ramanujan vector bundles.
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