On the Limit of the Tridiagonal Model for $\beta$-Dyson Brownian Motion

TL;DR

This paper investigates the asymptotic behavior of the Householder tridiagonalization applied to $eta$-Dyson Brownian motion, proposing a limit for the top-left minors and conjecturing a related stochastic operator for all $eta>0$.

Contribution

It extends the tridiagonal model analysis to $eta$-Dyson Brownian motion, providing explicit limits for fixed minors and conjecturing a new dynamical $eta$-stochastic Airy operator.

Findings

Proved the limit for $eta=1$.

Numerical evidence for $eta=1,2,4$.

Conjectured a dynamical $eta$-stochastic Airy operator.

Abstract

In previous work, a description of the result of applying the Householder tridiagonalization algorithm to a G$\beta$E random matrix is provided by Edelman and Dumitriu. The resulting tridiagonal ensemble makes sense for all $\beta>0$, and has spectrum given by the $\beta$-ensemble for all $\beta>0$. Moreover, the tridiagonal model has useful stochastic operator limits which was introduced and analyzed in subsequent studies. In this work, we analogously study the result of applying the Householder tridiagonalization algorithm to a G$\beta$E process which has eigenvalues governed by $\beta$-Dyson Brownian motion. We propose an explicit limit of the upper left $k \times k$ minor of the $n \times n$ tridiagonal process as $n \to \infty$ and $k$ remains fixed. We prove the result for $\beta=1$, and also provide numerical evidence for $\beta=1,2,4$. This leads us to conjecture the form of a dynamical $\beta$-stochastic Airy operator with smallest $k$ eigenvalues evolving according to the $n \to \infty$ limit of the largest, centered and re-scaled, $k$ eigenvalues of $\beta$-Dyson Brownian motion.