• Corpus ID: 241032926

Truncations of random symplectic unitary matrices

  title={Truncations of random symplectic unitary matrices},
  author={Boris A. Khoruzhenko and Serhii Lysychkin},
This paper is concerned with complex eigenvalues of truncated unitary quaternion matrices equipped with the Haar measure. The joint eigenvalue probability density function is obtained for truncations of any size. We also obtain the spectral density and the eigenvalue correlation functions in various scaling limits. In the limit of strong non-unitarity the universal complex Ginibre form of the correlation functions is recovered in the spectral bulk off the real line after unfolding the spectrum… 

Figures from this paper

Characteristic polynomials of random truncations: moments, duality and asymptotics

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in

Universal Scaling Limits of the Symplectic Elliptic Ginibre Ensemble

. We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite

Spherical induced ensembles with symplectic symmetry

. We consider the complex eigenvalues of the induced spherical Ginibre ensemble with symplectic symmetry and establish the local universality of these point processes along the real axis. We derive

Wronskian structures of planar symplectic ensembles

We consider the eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble, which are known to form a Pfaffian point process in the plane. It was recently

Phase transition of eigenvalues in deformed Ginibre ensembles

Consider a random matrix of size N as an additive deformation of the complex Ginibre ensemble under a deterministic matrix X 0 with a finite rank, independent of N . When some eigenvalues of X 0

Random normal matrices in the almost-circular regime

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow “band” around the unit circle of width proportional to 1 n , where n is the size of matrices. For general

On the almost‐circular symplectic induced Ginibre ensemble

We consider the symplectic‐induced Ginibre process, which is a Pfaffian point process on the plane. Let N be the number of points. We focus on the almost‐circular regime where most of the points lie

Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

An exact map was established by Lacroix-A-Chez-Toine et al. in (Phys Rev A 99(2):021602, 2019) between the N complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble,

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

This work considers random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfa-based structures, respectively, and derives the asymptotic expansions of the log-partition functions up to and including the O (1) -terms as the number N of particles increases.



Truncations of random unitary matrices

We analyse properties of non-Hermitian matrices of size M constructed as square submatrices of unitary (orthogonal) random matrices of size N >M , distributed according to the Haar measure. In this

Universality for products of random matrices I: Ginibre and truncated unitary cases

Recently, the joint probability density functions of complex eigenvalues for products of independent complex Ginibre matrices have been explicitly derived as determinantal point processes. We express

Universal microscopic correlation functions for products of truncated unitary matrices

We investigate the spectral properties of the product of M complex non-Hermitian random matrices that are obtained by removing L rows and columns of larger unitary random matrices uniformly

Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.

  • J. R. IpsenM. Kieburg
  • Mathematics, Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2014
This work proves a weak commutation relation of the random matrices at finite matrix sizes, which previously has been discussed for infinite matrix size, and derives the joint probability densities of the eigenvalues.

Fermionic mapping for eigenvalue correlation functions of weakly non-Hermitian symplectic ensemble

Products of independent quaternion Ginibre matrices and their correlation functions

We discuss the product of independent induced quaternion (β = 4) Ginibre matrices, and the eigenvalue correlations of this product matrix. The joint probability density function for the eigenvalues

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

This paper shows that a real n×n matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has 2nπ−−√+o(n√) real eigenvalues asymptotically almost surely.

Gap probabilities in non-Hermitian random matrix theory

We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles

Truncations of random orthogonal matrices.

An exact formula for the density of eigenvalues is derived which consists of two components: finite fraction ofeigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis.

Universal Scaling Limits of the Symplectic Elliptic Ginibre Ensemble

. We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite