论文标题
关于与beta集合有关的某些矩阵的平均密度，并应用于Toda晶格
On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice
论文作者
论文摘要
在本手稿中，我们研究了与经典$β$ imembles（高斯，拉瓜尔，雅各布，雅各布）相关的三角形随机矩阵模型，即矩阵的尺寸$ n $时，矩阵的大小$ n倾向于无限属于$βn=2α$ consent的约束，$βn$ constant，$ constant，$ n $ $ a $α> 0 $ 0。我们称这些合奏称为高斯，拉瓜尔和雅各比$α$增强物，我们证明了它们的经验频谱分布与状态平均密度的融合，我们明确地计算了它们。作为一种应用，我们明确计算了Toda晶格的LAX矩阵状态的平均密度，并具有有关Gibbs集合的周期性边界条件。
In this manuscript we study tridiagonal random matrix models related to the classical $β$-ensembles (Gaussian, Laguerre, Jacobi) in the high temperature regime, i.e. when the size $N$ of the matrix tends to infinity with the constraint that $βN=2α$ constant, $α> 0$. We call these ensembles the Gaussian, Laguerre and Jacobi $α$-ensembles and we prove the convergence of their empirical spectral distributions to their mean densities of states and we compute them explicitly. As an application we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.
