关于与β系综相关的某些矩阵的平均态密度及在Toda晶格中的应用
On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice
AI总结：
本文研究高温区域β系综相关三对角随机矩阵，证明经验谱分布收敛并显式计算平均态密度，应用于Toda晶格Lax矩阵。
摘要：
本文研究了与经典β系综（高斯、拉盖尔、雅可比）相关的三对角随机矩阵模型在高温区域，即当矩阵大小N趋于无穷且满足βN=2α（α>0为常数）时的性质。我们将这些系综称为高斯、拉盖尔和雅可比α系综，并证明了其经验谱分布收敛到平均态密度，且显式计算了这些平均态密度。作为应用，我们显式计算了具有周期边界条件的Toda晶格的Lax矩阵关于吉布斯系综的平均态密度。
英文摘要：
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.
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