Tracy–Widom method for Jánossy density and joint distribution of extremal eigenvalues of random matrices
نویسندگان
چکیده
Abstract The Jánossy density for a determinantal point process is the probability that an interval $I$ contains exactly $p$ points except those at $k$ designated loci. associated with integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ shown to be expressed as Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of transformed $\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$. We observe $\tilde{\mathbf{K}}$ satisfies Tracy and Widom’s criteria if $\mathbf{K}$ does, because structure map $(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application Tracy–Widom method [7] densities, in terms solution system differential equations endpoints interval. Our approach does not explicitly refer isomonodromic systems Painlevé employed preceding works. As illustrative examples we compute densities $k=1, p=0$ Airy Bessel kernels, related joint distributions two largest eigenvalues random Hermitian matrices smallest singular values complex matrices.
منابع مشابه
The Differential Entropy of the Joint Distribution of Eigenvalues of Random Density Matrices
Laizhen Luo 1,2, Jiamei Wang 3,*, Lin Zhang 4 and Shifang Zhang 5 1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China 2 School of Applied Science, Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China; [email protected] 3 Department of Mathematics, Anhui University of Technology, Ma’Anshan 243032, China 4 Institute of Math...
متن کاملDensity of Eigenvalues of Random Normal Matrices
The relation between random normal matrices and conformal mappings discovered by Wiegmann and Zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set. It is shown that for a suitable class of potentials the asymptotic density of eigenvalues is uniform with support in the interior domain of a simple smooth curve.
متن کاملA mathematically simple method based on denition for computing eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices
In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...
متن کاملAsymptotic Distribution of Eigenvalues and Degeneration of Sparse Random Matrices
This work is concerned with an asymptotical distribution of eigenvalues of sparse random matrices. It is shown that the semicircle law which is known for random matrices is also valid for the sparse random matrices with sparsity nIN=o(1), where n is the matrix size and 2N the number of non-zero elements of the matrix. The degree of degeneration is also estimated for the matrices with 2N—cn (c>0...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Progress of theoretical and experimental physics
سال: 2021
ISSN: ['1347-4081', '0033-068X']
DOI: https://doi.org/10.1093/ptep/ptab123