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 (Ann. Probab. 46 (2018) 2287-2344), and later extended to rank-one additive perturbations by the author and Shkolnikov (Ann. Inst. Henri Poincare Probab. Stat. 55 (2019) 1402-1438). Our method is applicable to a significantly more general class of random tridiagonal matrices than that considered in (Ann. Inst. Henri Poincare Probab. Stat. 55 (2019) 1402-1438; Ann. Probab. 46 (2018) 2287-2344), including some non-symmetric cases that are not covered by the stochastic operator formalism of Bloemendal, Ramirez, Rider, and Virag (Probab. Theory Related Fields 156 (2013) 795-825; J. Amer. Math. Soc. 24 (2011) 919-944). 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 Schrodinger operator with Gaussian white noise potential.