We investigate sampling \(\beta \)-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu and Edelman (J Math Phys 43(11):5830–5847, 2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated with the three classical Hermite, Laguerre, and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of more general \(\beta \)-ensembles and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (Commun Pure Appl Math 69(1): 145–199, 2016), that the Gibbs chain on Jacobi matrices of size N mixes in \(\mathcal {O}(\log N)\).

我们调查采样\（\ beta \）-集合的时间复杂度小于集合基数的三次方。遵循Dumitriu和Edelman（J Math Phys 43（11）：5830-5847，2002），我们将集合视为随机三对角矩阵（即随机Jacobi矩阵）的特征值。首先，我们对与三个经典Hermite，Laguerre和Jacobi乐团相关的三对角线模型进行统一和基本的处理。为此，我们在定义三对角矩阵的系数的连续重新参数化之间使用变量的简单更改。其次，我们推导一个近似采样器，用于模拟更通用的\（\ beta \）-集合并说明多项式势能有多快。该方法将Jacobi矩阵上的Gibbs采样器与这些矩阵的对角线结合在一起。实际上，即使对于大型合奏，也只有少数吉布斯通过才能满足特征值的边际分布以适合预期的理论分布。当可以精确地模拟Gibbs采样器中的条件时，对于最大特征值的波动，可以观察到相同的快速经验收敛。我们的实验结果支持Krishnapur等人的推测。（Commun Pure Appl Math 69（1）：145–199，2016），大小为N的Jacobi矩阵上的Gibbs链在\（\ mathcal {O}（\ log N）\）中混合。