Airy-kernel determinant on two large intervals

Asymptotics of the Airy-kernel determinant

The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.

Determinants of Airy Operators and Applications to Random Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of certain operators that are analogues of Wiener-Hopf operators. The determinant formulas yield information about the distribution functions for certain random variables that arise in random matrix theory when one rescales at the edge of the spectrum.

Sine-kernel determinant on two large intervals

We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.

Some computable Wiener-Hopf determinants and polynomials orthogonal on an arc of the unit circle

Some Wiener--Hopf determinants on [0,s] are calculated explicitly for all s>0. Their symbols are zero on an interval and they are related to the determinant with the sine-kernel appearing in the random matrix theory. The determinants are calculated by taking limits of Toeplitz determinants, which in turn are found from the related systems of polynomials orthogonal on an arc of the unit circle. As is known, the latter polynomials are connected to those orthogonal on an interval of the real axis. This connection is somewhat extended here. The determinants we compute originate from the Bernstein-…

On the Determinant of a Certain Wiener-Hopf + Hankel Operator

We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.

实线上矩阵值核的数值Fredholm行列式

Numerical Fredholm determinants for matrix-valued kernels on the real line

本研究拓展了Bornemann的分析，提出一种计算实线上矩阵值核定义的迹类、希尔伯特-施密特积分算子Fredholm行列式的数值方法，推导了截断与求积误差估计，并在Birman-Schwinger算子上验证了效果。