On the Convergence of Random Tridiagonal Matrices to Stochastic Semigroups

We develop an improved version of the stochastic semigroup approach to study the edge of $\beta$-ensembles pioneered by Gorin and Shkolnikov, and later extended to rank-one additive perturbations by the author and Shkolnikov. Our method is applicable to a significantly more general class of random tridiagonal matrices than that considered in these previous works, including some non-symmetric cases that are not covered by the stochastic operator formalism of Bloemendal, Ram\'irez, Rider, and Vir\'ag. We present two applications of our main results: Firstly, we prove the convergence of $\beta$-Laguerre-type (i.e., sample covariance) random tridiagonal matrices to the stochastic Airy semigroup and its rank-one spiked version. Secondly, we prove the convergence of the eigenvalues of a certain class of non-symmetric random tridiagonal matrices to the spectrum of a continuum Schr\"odinger operator with Gaussian white noise potential.