The bead process introduced by Boutillier is a countable interlacing of the \({\text {Sine}}_2\) point processes. We construct the bead process for general \({\text {Sine}}_{\beta }\) processes as an infinite dimensional Markov chain whose transition mechanism is explicitly described. We show that this process is the microscopic scaling limit in the bulk of the Hermite \(\beta \) corner process introduced by Gorin and Shkolnikov, generalizing the process of the minors of the Gaussian Unitary and Orthogonal Ensembles. In order to prove our results, we use bounds on the variance of the point counting of the circular and the Gaussian beta ensembles, proven in a companion paper (Najnudel and Virág in Some estimates on the point counting of the Circular and the Gaussian Beta Ensemble, 2019).

Boutillier 引入的珠过程是\({\text {Sine}}_2\)点过程的可数交错。我们将一般\({\text {Sine}}_{\beta }\)过程的珠子过程构造为无限维马尔可夫链，其转换机制被明确描述。我们表明这个过程是 Hermite \(\beta \)Gorin 和 Shkolnikov 引入的角过程，概括了高斯幺正和正交系综的未成年人的过程。为了证明我们的结果，我们使用了圆和高斯 Beta 系综的点计数方差的界限，这在一篇配套论文中得到了证明（Najnudel 和 Virág 在《关于圆和高斯 Beta 系综的点计数的一些估计》中） , 2019)。