Harer-Zagier type recursion formula for the elliptic GinOE

We consider real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter $\tau \in [0,1]$, and present a Harer-Zagier type recursion formula for the even moments in the form of an $11$-term recurrence relation. For the Ginibre case when $\tau=0$, this formula simplifies to a 3-term recurrence relation. On the other hand, for the GOE case when $\tau=1$, it reduces to a 5-term recurrence relation, recovering the result established by Ledoux. For the proof, we employ the skew-orthogonal polynomial formalism and the generalised Christoffel-Darboux formula. Together with Gaussian integration by parts, these enable us to derive a seventh-order linear differential equation for the moment generating function.Comment: 36 pages, 1 figur

Determinantal Coulomb gas ensembles with a class of discrete rotational symmetric potentials

We consider determinantal Coulomb gas ensembles with a class of discrete rotational symmetric potentials whose droplets consist of several disconnected components. Under the insertion of a point charge at the origin, we derive the asymptotic behaviour of the correlation kernels both in the macro- and microscopic scales. In the macroscopic scale, this particularly shows that there are strong correlations among the particles on the boundary of the droplets. In the microscopic scale, this establishes the edge universality. For the proofs, we use the nonlinear steepest descent method on the matrix Riemann-Hilbert problem to derive the asymptotic behaviours of the associated planar orthogonal polynomials and their norms up to the first subleading terms.Comment: 25 pages, 5 figure

Scaling limits of complex and symplectic non-Hermitian Wishart ensembles

Non-Hermitian Wishart matrices were introduced in the context of quantum chromodynamics with a baryon chemical potential. These provide chiral extensions of the elliptic Ginibre ensembles as well as non-Hermitian extensions of the classical Wishart/Laguerre ensembles. In this work, we investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry classes of complex and symplectic Ginibre ensembles. We introduce a generalised Christoffel-Darboux formula in the form of a certain second-order differential equation, offering a unified and robust method for analyzing correlation functions across all scaling regimes in the model. By employing this method, we derive universal bulk and edge scaling limits for eigenvalue correlations at both strong and weak non-Hermiticity.Comment: 34 pages, 3 figure

Almost-Hermitian random matrices and bandlimited point processes

We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow "band" about the real line, of height proportional to $\tfrac 1 N$, where $N$ denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Ward's equation and a property which we call "cross-section convergence", which relates the large-$N$ limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko-Pastur law for LUE.Comment: 42 pages, 10 figure

The comparison of the aid allocation of Korea in the 1950s and the 1960s to that of contemporary Uganda

노트 : Prepared for Korea and World Economy Conference X

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 scaling limits of all correlation functions at regular points both in the strong and weak non-unitary regimes as well as at the origin having spectral singularity. A key ingredient of our proof is a derivation of a differential equation satisfied by the correlation kernels of the associated Pfaffian point processes, thereby allowing us to perform asymptotic analysis

Supplementary education at college and its consequences for individuals\u27 labor market outcomes in the United States

The current study seeks to expand our knowledge on extended education and its potential contribution to social inequality by examining socioeconomic disparities in supplementary education (SE) at college and its impact on labor market outcomes. Using data from the United States Education Longitudinal Study, logistic and linear regressions deliver the following main findings: (1) Socioeconomic status (SES) significantly affects SE participation, net of other factors. (2) With higher involvement in SE activities, neither employment nor income prospects significantly increase. (3) Low SES graduates are slightly more likely to benefit from SE than high SES graduates. (4) Among high-impact SE practices, only internships exert a positive effect on labor market outcomes. (DIPF/Orig.

Lemniscate ensembles with spectral singularity

We consider a family of random normal matrix models whose eigenvalues tend to occupy lemniscate type droplets as the size of the matrix increases. Under the insertion of a point charge, we derive the scaling limit at the singular boundary point, which is expressed in terms of the solution to the model Painlev\'{e} IV Riemann-Hilbert problem. For this, we apply a version of the Christoffel-Darboux identity and the strong asymptotics of the associated orthogonal polynomials, where the latter was obtained by Bertola, Elias Rebelo, and Grava.Comment: 29 pages, 5 figure