Determinants of Airy Operators and Applications to Random Matrices

Large gap asymptotics for random matrices

We outline an approach recently used to prove formulae for the multiplicative constants in the asymptotics for the sine-kernel and Airy-kernel determinants appearing in random matrix theory and related areas.

有限Toeplitz加Hankel矩阵具有有理符号的行列式公式

Determinant formulas for finite Toeplitz plus Hankel matrices with rational symbols

本文将Day公式推广至有理符号的有限Toeplitz加Hankel矩阵行列式，通过Fredholm行列式给出精确表达式，并用于分析极限特征值。

Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.

Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matrices

We compute the asymptotics of the determinants of certain $n\times n$ Toeplitz + Hankel matrices $T_n(a)+H_n(b)$ as $n\to\infty$ with symbols of Fisher-Hartwig type. More specifically we consider the case where $a$ has zeros and poles and where $b$ is related to $a$ in specific ways. Previous results of Deift, Its and Krasovsky dealt with the case where $a$ is even. We are generalizing this in a mild way to certain non-even symbols.

Trace and determinant preserving maps of matrices

Suppose a map $ϕ$ on the set of positive definite matrices satisfies $\det(A+B)=\det(ϕ(A)+ϕ(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(ϕ(A){ϕ(B)}^{-1}).$$ Through this viewpoint, we show that $ϕ$ is of the form $ϕ(A)= M^*AM$ or $ϕ(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $ϕ: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.

Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results

We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.