Reed College · B.A. Mathematics · Senior Thesis · Math 470 · 2021 · Solo thesis · advised by Nate Wells
Spectral Statistics of Random Matrices
A 109-page honors thesis on the eigenvalue distributions of classical random-matrix ensembles, paired with RMAT — an R package I wrote to simulate the ensembles and study their spectra.
My senior thesis studies random matrices — matrices whose entries are drawn at random — and the statistics of their eigenvalues. As the matrices grow, their eigenvalues settle into strikingly predictable shapes (Wigner’s semicircle law is the classic example). The thesis surveys these results across the Gaussian, Wishart, and Hermite β-ensembles, with the Dyson index tying the families together.
Alongside the exposition I wrote RMAT, an R package that generates each ensemble, computes its spectrum, and visualizes two statistics: the distribution of the eigenvalues themselves and the spacings between them. The package handles the matrix generation, spectral computation, plotting, and parallelization used throughout the thesis.
The result is a 109-page document that pairs rigorous mathematical exposition — with appendices on linear algebra, probability, and Markov chains — against a reusable computational toolkit. It’s the most sustained piece of mathematical work I’ve done, and the closest in spirit to the probability that sits under actuarial practice.