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.

翻译：在先前的工作中，Edelman与Dumitriu描述了将Householder三对角化算法应用于G$\beta$E随机矩阵的结果。所得的三对角系综对所有$\beta>0$均有定义，且其谱由$\beta$-系综给出（对所有$\beta>0$）。此外，该三对角模型具有有用的随机算子极限，这些极限已在后续研究中被引入和分析。本文中，我们类比研究了将Householder三对角化算法应用于G$\beta$E过程的结果，该过程的特征值由$\beta$-Dyson布朗运动所支配。我们提出了当$n \to \infty$且$k$固定时，$n \times n$三对角过程左上角$k \times k$子矩阵的一个显式极限。我们证明了$\beta=1$时的结果，并为$\beta=1,2,4$提供了数值证据。这引导我们推测一个动态$\beta$-随机Airy算子的形式，其最小的$k$个特征值按照$\beta$-Dyson布朗运动最大、中心化并重新标度的$k$个特征值在$n \to \infty$时的极限演化。