In previous work, Edelman and Dumitriu provide a description of the result of applying the Householder tridiagonalization algorithm to a G$\beta$E random matrix. 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 introduced and analyzed by Edelman and Sutton, and subsequently analyzed in work by Ramirez, Rider, and Vir\'ag. 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.

翻译：在先前的研究中，Edelman与Dumitriu描述了将Householder三对角化算法应用于GβE随机矩阵所得到的结果。所得的三对角系综对所有β>0均有定义，其谱由β-系综给出（对所有β>0成立）。此外，该三对角模型具有由Edelman和Sutton引入并分析、后经Ramirez、Rider和Virág的工作进一步研究的随机算子极限。本文中，我们类比研究了将Householder三对角化算法应用于GβE过程的结果，该过程的特征值由β-戴森布朗运动所支配。我们提出了当n→∞且k固定时，n×n三对角过程左上角k×k子矩阵的显式极限。我们证明了β=1情形下的结果，并为β=1,2,4提供了数值证据。这促使我们推测一个动态β-随机Airy算子的形式，该算子的最小k个特征值按照β-戴森布朗运动最大k个特征值经中心化与重标度后的n→∞极限演化。