arXiv subjects

Sung-Soo Byun

Publications and source records attributed to Sung-Soo Byun.

At least 37 records · Page 2Linked to original sources

arXiv2025

Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE

We consider a planar Coulomb gas ensemble of size $N$ with the inverse temperature $β=2$ and external potential $Q(z)=|z|^2-2c \log|z-a|$, where $c>0$ and $a \in \mathbb{C}$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1) N \times (c+1) N$ conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$. Depending on the values of $c$ and $a$, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-$N$ expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our proof is based on a refined Riemann-Hilbert analysis for planar orthogonal polynomials using the partial Schlesinger transform.

math-ph↗

arXiv2025

Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble

We study the integrable structure and scaling limits of the conditioned eigenvector overlap of the symplectic Ginibre ensemble of Gaussian non-Hermitian random matrices with independent quaternion elements. The average of the overlap matrix elements constructed from left and right eigenvectors, conditioned to $x$, are derived in terms of a Pfaffian determinant. Regarded as a two-dimensional Coulomb gas with the Neumann boundary condition along the real axis, it contains a kernel of skew-orthogonal polynomials with respect to the weight function $ω^{(\mathrm{over})}(z)=|z-\overline{x}|^2(1+|z-x|^2)e^{-2|z|^2}$, including a non-trivial insertion of a point charge. The mean off-diagonal overlap is related to the diagonal (self-)overlap by a transposition, in analogy to the complex Ginibre ensemble. For $x$ conditioned to the real line, extending previous results at $x=0$, we determine the skew-orthogonal polynomials and their skew-kernel with respect to $ω^{(\mathrm{over})}(z)$. This is done in two steps and involves a Christoffel perturbation of the weight $ω^{(\mathrm{over})}(z)=|z-\overline{x}|^2ω^{(\mathrm{pre})}(z)$, by computing first the corresponding quantities for the unperturbed weight $ω^{(\mathrm{pre})}(z)$. Its kernel is shown to satisfy a differential equation at finite matrix size $N$. This allows us to take different large-$N$ limits, where we distinguish bulk and edge regime along the real axis. The limiting mean diagonal overlaps and corresponding eigenvalue correlation functions of the point processes with respect to $ω^{(\mathrm{over})}(z)$ are determined. We also examine the effect on the planar orthogonal polynomials when changing the variance in $ω^{(\mathrm{pre})}(z)$, as this appears in the eigenvector statistics of the complex Ginibre ensemble.

math-ph↗

arXiv2025

Real Eigenvalues of Asymmetric Wishart Matrices: Expected Number, Global Density and Integrable Structure

We investigate the real eigenvalues of asymmetric Wishart matrices of size $N$, indexed by the rectangular parameter $ν\in \mathbb{N}$ and the non-Hermiticity parameter $τ\in [0,1]$. The rectangular parameter $ν$ is either fixed or proportional to $N$. The non-Hermiticity parameter $τ$ is either fixed or $τ= 1 - O(1/N)$, corresponding to the strongly and weakly non-Hermitian regimes, respectively. We establish a decomposition structure for the finite-$N$ correlation kernel of the real eigenvalues, which form Pfaffian point processes. Taking the symmetric limit $τ= 1$, where the model reduces to the Laguerre orthogonal ensemble, this decomposition structure reduces to the known rank-one perturbation structure established by Adler, Forrester, Nagao, and van Moerbeke, as well as by Widom. Using the decomposition structure, we show that the expected number of real eigenvalues is proportional to $\sqrt{N}$ in the strongly non-Hermitian regime and to $N$ in the weakly non-Hermitian regime, providing explicit leading coefficients in both cases. Furthermore, we derive the limiting real eigenvalue densities, which recovers the Marchenko-Pastur distribution in the symmetric limit.

math.PR↗

arXiv2025

Free energy of spherical Coulomb gases with point charges

We consider two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials that are invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models can be interpreted as Coulomb gases on the complex plane with weakly confining potentials, where the associated droplet is the entire complex plane. For these models, we derive precise asymptotic expansions of the free energies, including the constant terms.

math-ph↗

arXiv2025

Properties of the one-component Coulomb gas on a sphere with two macroscopic external charges

The one-component Coulomb gas on the sphere, consisting on $N$ unit charges interacting via a logarithmic potential, and in the presence of two external charges each of strength proportional to $N$, is considered. There are two spherical caps naturally associated with the external charges, giving rise to two distinct phases depending on them not overlapping (post-critical) or overlapping (pre-critical). The equilibrium measure in the post-critical phase is known from earlier work. We determine the equilibrium measure in the pre-critical phase using a particular conformal map, with the parameters therein specified in terms of a root of a certain fourth order polynomial. This is used to determine the exact form of the electrostatic energy for the pre-critical phase. Using a duality relation from random matrix theory, the partition function for the Coulomb gas at the inverse temperature $β= 2$ can be expanded for large $N$ in the post-critical phase, and in a scaling region of the post and pre-critical boundary. For the pre-critical phase, the duality identity implies a relation between two electrostatic energies, one for the present sphere system, and the other for a certain constrained log-gas relating to the Jacobi unitary ensemble.

math-ph↗

arXiv2024

Conformal field theory of Gaussian free fields in a multiply connected domain

We implement a version of conformal field theory (CFT) that gives a connection to SLE in a multiply connected domain. Our approach is based on the Gaussian free field and applies to CFTs with central charge $c \leq 1$. In this framework we introduce the generalized Eguchi-Ooguri equations and use them to derive the explicit form of Ward's equations, which describe the insertion of a stress tensor in terms of Lie derivatives and differential operators depending on the Teicmüller modular parameters. Furthermore, by implementing the BPZ equations, we provide a conformal field theoretic realization of an SLE in a multiply connected domain, which in particular suggests its drift function, and construct a class of martingale observables for this SLE process.

math-ph↗

arXiv2024

Harer-Zagier type recursion formula for the elliptic GinOE

We consider the real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter $τ\in [0,1]$, and present a Harer-Zagier type recursion formula for the even moments in the form of an $11$-term recurrence relation. For the symmetric GOE case ($τ=1$), it reduces to a known 5-term recurrence relation. On the other hand, for the asymmetric cases when $τ< 1$, the recursion formula is new, even in the special case of the well-studied Ginibre ensemble ($τ=0$), where it reduces to a 3-term recurrence. For the proof, we derive a seventh-order linear differential equation for the moment generating function.

math.PR↗

arXiv2024

Large gap probabilities of complex and symplectic spherical ensembles with point charges

We consider $n$ eigenvalues of complex and symplectic induced spherical ensembles, which can be realised as two-dimensional determinantal and Pfaffian Coulomb gases on the Riemann sphere under the insertion of point charges. For both cases, we show that the probability that there are no eigenvalues in a spherical cap around the poles has an asymptotic behaviour as $n\to \infty$ of the form $$ \exp\Big( c_1 n^2 + c_2 n\log n + c_3 n + c_4 \sqrt n + c_5 \log n + c_6 + \mathcal{O}(n^{-\frac1{12}}) \Big) $$ and determine the coefficients explicitly. Our results provide the second example of precise (up to and including the constant term) large gap asymptotic behaviours for two-dimensional point processes, following a recent breakthrough by Charlier.

math-ph↗

arXiv2024

Spectral Moments of the real Ginibre ensemble

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large $N$ expansions. These topics are inter-related. For example, a third order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second order difference equation for the general complex moments $M_{2p}^{\rm r}$. The latter are expressed in terms of the ${}_3 F_2$ hypergeometric functions, with a simplification to the ${}_2 F_1$ hypergeometric function possible for $p=0$ and $p=1$, allowing for the large $N$ expansion of these moments to be obtained. The large $N$ expansion involves both integer and half integer powers of $1/N$. The three term recurrence then provides the large $N$ expansion of the full sequence $\{ M_{2p}^{\rm r} \}_{p=0}^\infty$. Fourth and third order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and properties of the large $N$ expansion of these quantities are determined.

math-ph↗

arXiv2024

$q$-deformed Gaussian unitary ensemble: spectral moments and genus-type expansions

The eigenvalue probability density function of the Gaussian unitary ensemble permits a $q$-extension related to the discrete $q$-Hermite weight and corresponding $q$-orthogonal polynomials. A combinatorial counting method is used to specify a positive sum formula for the spectral moments of this model. The leading two terms of the scaled $1/N^2$ genus-type expansion of the moments are evaluated explicitly in terms of the incomplete beta function. Knowledge of these functional forms allows for the smoothed leading eigenvalue density and its first correction to be determined analytically.

math.PR↗

arXiv2024

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.

math.PR↗

arXiv2023

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

In this article, we compute and compare the statistics of the number of eigenvalues in a centred disc of radius $R$ in all three Ginibre ensembles. We determine the mean and variance as functions of $R$ in the vicinity of the origin, where the real and symplectic ensembles exhibit respectively an additional attraction to or repulsion from the real axis, leading to different results. In the large radius limit, all three ensembles coincide and display a universal bulk behaviour of $O(R^2)$ for the mean, and $O(R)$ for the variance. We present detailed conjectures for the bulk and edge scaling behaviours of the real Ginibre ensemble, having real and complex eigenvalues. For the symplectic ensemble we can go beyond the Gaussian case (corresponding to the Ginibre ensemble) and prove the universality of the full counting statistics both in the bulk and at the edge of the spectrum for rotationally invariant potentials, extending a recent work which considered the mean and the variance. This statistical behaviour coincides with the universality class of the complex Ginibre ensemble, which has been shown to be associated with the ground state of non-interacting fermions in a two-dimensional rotating harmonic trap. All our analytical results and conjectures are corroborated by numerical simulations.

math-ph↗

arXiv2023

Finite size corrections for real eigenvalues of the elliptic Ginibre matrices

We consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.

math.PR↗

arXiv2023

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.

math.PR↗

arXiv2023

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. As an application of our approach, we prove the bulk universality of the almost-circular ensembles.

math.PR↗

arXiv2023

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 Painlevé IV Riemann-Hilbert problem. For this, we introduce a version of the Christoffel-Darboux identity and combine it with the strong asymptotics of the associated orthogonal polynomials due to Bertola, Elias Rebelo and Grava.

math.PR↗

arXiv2023

Large deviations and fluctuations of real eigenvalues of elliptic random matrices

We study real eigenvalues of $N\times N$ real elliptic Ginibre matrices indexed by a non-Hermiticity parameter $0\leq τ<1$, in both the strong and weak non-Hermiticity regime. Here $N$ is assumed to be an even number. In both regimes, we prove a central limit theorem for the number of real eigenvalues. We also find the asymptotic behaviour of the probability $p_{N,k}^{(τ)}$ that exactly $k$ eigenvalues are real. In the strong non-Hermiticity regime, where $τ$ is fixed, we find \begin{align*} \lim_{N\to\infty} \frac{1}{\sqrt{N}} \log p_{N,k_N}^{(τ)} = -\sqrt\frac{1+τ}{1-τ} \frac{ζ(3/2)}{\sqrt{2π}} \end{align*} for any sequence $(k_N)_N$ of even numbers such that $k_N = o(\frac{\sqrt N}{\log N})$ as $N\to\infty$, where $ζ$ is the Riemann zeta function. In the weak non-Hermiticity regime, where $τ=1-\frac{α^2}{N}$, we obtain \begin{align*} \lim_{N\to\infty} \frac{1}{N} \log p_{N,k_N}^{(τ)} \leq \frac{2}π \int_0^1 \log\left(1-e^{-α^2 s^2}\right) \sqrt{1-s^2} \, ds \end{align*} for any sequence $(k_N)_N$ of even numbers such that $k_N=o(\frac{N}{\log N})$ as $n\to\infty$. This inequality is expected to be an equality.

math-ph↗

arXiv2023

Progress on the study of the Ginibre ensembles I: GinUE

The Ginibre unitary ensemble (GinUE) consists of $N \times N$ random matrices with independent complex standard Gaussian entries. This was introduced in 1965 by Ginbre, who showed that the eigenvalues form a determinantal point process with an explicit correlation kernel, and after scaling they are supported on the unit disk with constant density. For some time now it has been appreciated that GinUE has a fundamental place within random matrix theory, both for its applications and for the richness of its theory. Here we review the progress on a number of themes relating to the study of GinUE. These are eigenvalue probability density functions and correlation functions, fluctuation formulas, sum rules and asymptotic behaviours of correlation functions, and normal matrix models. We discuss too applications in quantum many body physics and quantum chaos, and give an account of some statistical properties of the eigenvectors.

math-ph↗