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Sung-Soo Byun

Publications and source records attributed to Sung-Soo Byun.

At least 19 recordsLinked to original sources
arXiv2026

Disc counting statistics of the real Ginibre ensemble

We study the disc counting statistics of the real Ginibre ensemble, whose spectrum consists of real and non-real eigenvalues, the latter occurring in complex-conjugate pairs. As the matrix dimension increases, we derive the asymptotic behaviour of the joint cumulants of the numbers of real and non-real eigenvalues contained in a centred disc in the bulk, edge, and exterior regimes of the circular law. As consequences, we establish a joint central limit theorem and obtain the conjectured asymptotics for the number variance. Furthermore, our results reveal a universality phenomenon: although the fluctuations of the real and non-real eigenvalues separately retain dependence on the symmetry class, their combined fluctuations exhibit the same limiting behaviour as in the previously established complex and symplectic Ginibre ensembles.

math.PR↗
arXiv2026

Confinement transitions in half-space constrained Riesz gases

We study a Riesz gas with interaction parameter $s \in (d-3,d)$ in an arbitrary spatial dimension $d$, confined to a half-space by a hard wall. We prove that the equilibrium measure exhibits a dichotomy according to the interaction range. For $s \in (d-2,d)$, corresponding to the weakly long-ranged regime, the equilibrium measure always retains a non-trivial bulk component and thus is never completely confined to the wall. In contrast, for $s \in (d-3,d-2]$, corresponding to the strongly long-ranged regime, a confinement transition occurs: there exists a critical wall position beyond which the equilibrium measure is supported entirely on the wall. Our theorem generalises recent results established for the Coulomb gas, corresponding to the case $s=d-2$.

math-ph↗
arXiv2026

Three topological phases of the elliptic Ginibre ensembles with a point charge

We consider the complex and symplectic elliptic Ginibre matrices of size $(c+1)N \times (c+1)N$, conditioned to have a deterministic eigenvalue at $ p \in \mathbb{R} $ with multiplicity $ c N $. We show that their limiting spectrum is either simply connected, doubly connected, or composed of two disjoint simply connected components. Moreover, denoting by $τ\in [0,1]$ the non-Hermiticity parameter, we explicitly characterise the regions in the parameter space $ (p, c, τ) $ where each topological type emerges. For cases where the droplet is either simply or doubly connected, we provide an explicit description of the limiting spectrum and the corresponding electrostatic energies. As an application, we derive the asymptotic behaviour of the moments of the characteristic polynomial for elliptic Ginibre matrices in the exponentially varying regime.

math-ph↗
arXiv2026

Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.

math.PR↗
arXiv2026

Free energy expansion of determinantal Coulomb gases in the quadratic fields with a point charge

We study a determinantal Coulomb gas in the complex plane associated with the external potential $$ Q(z)=\frac{1}{1-τ^2}\big(|z|^2-τ\text{Re } z^2\big)-2c\log|z-a|, $$ where $τ\in[0,1)$, $c\ge0$, and $a\ge0$. In the regimes where the associated droplet is simply or doubly connected, we derive the free energy expansion up to and including the constant term, with all coefficients computed explicitly, thereby extending recent results in the isotropic case $τ=0$. In particular, we identify the constant term with the Liouville action associated with the droplet. Our result admits a natural interpretation in terms of asymptotic expansions of moments of characteristic polynomials for the elliptic Ginibre ensemble. The proof is based on a deformation framework involving both the singularity location $a$ and the anisotropy parameter $τ$, relating variations of the free energy to refined asymptotics of planar orthogonal polynomials. The asymptotic analysis relies on the foliation flow method of Hedenmalm and Wennman, providing an alternative to the Riemann--Hilbert approach used in the isotropic setting. The present work suggests a general framework connecting free energy expansions, refined asymptotics of planar orthogonal polynomials, and conformally invariant geometric functionals, with several intermediate results already formulated for general algebraic Hele-Shaw potentials.

math-ph↗
arXiv2026

Dissipative Spectral Form Factor of the Complex Elliptic Ginibre Ensemble across Various Non-Hermiticity Regimes

We study the dissipative spectral form factor (DSFF) at complex time $T e^{iθ}$ for the complex elliptic Ginibre ensemble with non-Hermiticity parameter $τ\in [0,1)$. As the matrix dimension $N \to \infty$, we consider the natural scalings in both the time variable and the non-Hermiticity parameter, namely $T = O(N^γ)$ and $1 - τ= O(N^{-α})$. For all regimes $γ\ge 0$ and $α\ge 0$, we derive the precise asymptotic behaviour of both the disconnected and connected components of the DSFF. In particular, we explicitly characterise the dip--ramp--plateau structure, including the dip time and the Heisenberg time. In addition, we identify the mesoscopic regime $α\in (0,1)$, which interpolates between the behaviour of the DSFF of non-Hermitian random matrices and the spectral form factor (SFF) of Hermitian ensembles. We further provide an explicit description of the phase diagram, in which the ramp exhibits quadratic, linear, or intermediate behaviour depending on the scaling parameters.

math-ph↗
arXiv2026

Spectral moments of complex and symplectic non-Hermitian random matrices

We study non-Hermitian random matrices belonging to the symmetry classes of the complex and symplectic Ginibre ensemble, and present a unifying and systematic framework for analysing mixed spectral moments involving both holomorphic and anti-holomorphic parts. For weight functions that induce a recurrence relation of the associated planar orthogonal polynomials, we derive explicit formulas for the spectral moments in terms of their orthogonal norms. This includes exactly solvable models such as the elliptic Ginibre ensemble and non-Hermitian Wishart matrices. In particular, we show that the holomorphic spectral moments of complex non-Hermitian random matrices coincide with those of their Hermitian limit up to a multiplicative constant, determined by the non-Hermiticity parameter. Moreover, we show that the spectral moments of the symplectic non-Hermitian ensemble admit a decomposition into two parts: one corresponding to the complex ensemble and the other constituting an explicit correction term. This structure closely parallels that found in the Hermitian setting, which naturally arises as the Hermitian limit of our results. Within this general framework, we perform a large-$N$ asymptotic analysis of the spectral moments for the elliptic Ginibre and non-Hermitian Wishart ensemble, revealing the mixed moments of the elliptic and non-Hermitian Marchenko--Pastur laws. Furthermore, for the elliptic Ginibre ensemble, we employ a recently developed differential operator method for the associated correlation kernel, to derive an alternative explicit formula for the spectral moments and obtain their genus-type large-$N$ expansion.

math-ph↗
arXiv2026

Numerical ranges of non-normal random matrices: elliptic Ginibre and non-Hermitian Wishart ensembles

The numerical range of a non-normal matrix plays a central role as a descriptor of non-normal effects beyond spectral information. We study a class of fundamental non-Hermitian random matrix ensembles that interpolate between the Hermitian and non-Hermitian regimes. Our analysis focuses on the elliptic Ginibre ensemble and its chiral counterpart, as well as on non-Hermitian Wishart matrices. For each of these models, we explicitly characterise the geometry of the numerical range in the large-system limit. In particular, we show that for the elliptic Ginibre ensemble and its chiral version, the limiting numerical range is an ellipse, whereas for the non-Hermitian Wishart ensemble it is described by a non-elliptic envelope. Furthermore, we determine the numerical range of products of $n$ independent elliptic Ginibre matrices, which recovers, in the cases $n=1$ and $n=2$, the results for the elliptic Ginibre ensemble and the non-Hermitian Wishart ensemble at maximal non-Hermiticity, respectively.

math.PR↗
arXiv2026

Upper tail large deviations for extremal eigenvalues of the real, complex and symplectic elliptic Ginibre matrices

We consider the elliptic Ginibre ensembles in the real, complex and symplectic symmetry classes. As the matrix size tends to infinity, we derive the asymptotic behaviour of the upper tail large deviation probabilities for both the spectral radius and the rightmost eigenvalue. More generally, we obtain asymptotic formulas for the probability that an eigenvalue is found in a prescribed region outside the support of the elliptic law, thereby providing a unified framework in which the results for the spectral radius and the rightmost eigenvalue appear as special cases. The key ingredient of our analysis is the precise asymptotic behaviour of the associated one-point functions, which is of independent interest.

math.PR↗
arXiv2026

Electrostatic computations for statistical mechanics and random matrix applications

Although for the most part classical, the topic of electrostatics finds to this day new applications. In this review we highlight several theoretical results on electrostatics, chosen to both illustrate general principles, and for their application in statistical mechanics and random matrix settings. The theoretical results include electrostatic potentials and energies associated with balls and hyperellipsoids in general dimension, equilibrium measures associated with surfaces of the latter, the use of conformal mappings in two-dimensions, and the balayage measure. A number of explicit examples of their use in predicting the leading asymptotic form of certain configuration integrals and particle density in particular statistical mechanical systems are given, as well as with regards to questions relating to fluctuation formulas and (conditioned) gap probabilities.

math-ph↗
arXiv2026

Equilibrium measures for higher dimensional rotationally symmetric Riesz gases

We study equilibrium measures for Riesz gases in dimension $d$ with pairwise interaction kernel $|x-y|^{-s}$, subject to radially symmetric external fields. We characterise broad classes of confining potentials for which the equilibrium measure is supported on the unit ball and admits an explicit density. Our main contribution is a converse construction: starting from a prescribed radially symmetric equilibrium density given as a power series in the squared radius, we determine the associated external potential and establish the corresponding Euler-Lagrange variational conditions. A key ingredient in the proof is an identity between two ${}_3F_2$ hypergeometric functions evaluated at unit argument, which is of independent interest. As applications, we identify the external potentials corresponding to equilibrium densities proportional to $(1-|x|^2)^α$, $α>-1$, and show that these potentials can be expressed in terms of Gauss hypergeometric functions ${}_2F_1$, reducing to polynomials for special values of $α$. We also determine the equilibrium measure associated with purely power-type external potentials, often referred to as Freud or Mittag--Leffler potentials in the context of log gases, for which the equilibrium density admits an explicit ${}_2F_1$ representation. Furthermore, we apply our framework to a Coulomb gas in dimension $d+1$ confined by a harmonic potential to the half-space. We derive a necessary condition under which the equilibrium measure is fully supported on the boundary hyperplane of dimension $d$, with the induced density corresponding to that of a Riesz gas with exponent $s=d-1$.

math-ph↗
arXiv2026

$q$-deformation of the Marchenko-Pastur law

We study a $q$-deformed random unitary ensemble associated with the little-$q$ Laguerre weight, which provides a discrete analogue of the classical Laguerre unitary ensemble. In the double scaling regime $q=e^{-λ/N}$, where $N$ is the system size and $λ\ge 0$, we derive the limiting spectral distribution as $N\to \infty$, which yields a $q$-deformation of the Marchenko-Pastur law. The limiting density undergoes a phase transition at an explicitly determined critical value $λ_c$: for $λ<λ_c$, the support consists of a single band region, whereas for $λ>λ_c$ an additional saturated region emerges adjacent to the band region. Our derivation of the limiting distribution is based on three complementary approaches: the method of moments, the analysis of a constrained equilibrium problem, and the asymptotic zero distribution of orthogonal polynomials. As a consequence, we establish the convergence of the empirical measure as well as a large deviation principle. In addition, we derive closed-form expressions for the spectral moments using the combinatorial structure of orthogonal polynomials, and obtain large-$N$ expansions for these moments.

math.PR↗
arXiv2025

The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble

We investigate real eigenvalues of real elliptic Ginibre matrices of size $n$, indexed by the parameter of asymmetry $τ\in [0,1]$. In both the strongly and weakly non-Hermitian regimes, where $τ\in [0,1)$ is fixed or $1-τ=O(1/n)$, respectively, we derive the asymptotic expansion of the probability $p_{n,n-2l}$ that all but a finite number $2l$ of eigenvalues are real. In particular, we show that the expansion is of the form \begin{align*} \log p_{n, n-2l} = \begin{cases} a_1 n^2 +a_2 n + a_3 \log n +O(1) &\text{at strong non-Hermiticity}, \\ b_1 n +b_2 \log n + b_3 +o(1) &\text{at weak non-Hermiticity}, \end{cases} \end{align*} and we determine all coefficients explicitly. Furthermore, in the special case where $l=1$, we derive the full-order expansions. For the proofs, we employ distinct methods for the strongly and weakly non-Hermitian regimes. In the former case, we utilise potential-theoretic techniques to analyse the free energy of elliptic Ginibre matrices conditioned to have $n-2l$ real eigenvalues, together with the strong Szegő limit theorems. In the latter case, we utilise the skew-orthogonal polynomial formalism and the asymptotic behaviour of the Hermite polynomials.

math.PR↗
arXiv2025

Precise large deviations in geometric last passage percolation

We study the last passage time in geometric last passage percolation (LPP). As the system size increases, we derive precise large deviation probabilities -- up to and including the constant terms -- for both the lower and upper tails. A key step in proving these results is to establish a duality formula that reformulates the LPP problem in terms of the largest eigenvalue in the Jacobi unitary ensemble (JUE). In addition, we establish a second duality formula, which relates the LPP problem to the truncated unitary ensemble (TUE). Using this, we also derive asymptotics for the moments of the absolute value of characteristic polynomials of the TUE, which may be of independent interest.

math.PR↗
arXiv2025

Anomalous free energy expansions of planar Coulomb gases: multi-component and conformal singularity

We study the partition function $$ Z_n = \int_{\mathbb{C}^n } \prod_{1 \le j -1$ and $$ V(z)= |z|^{2d}-t(z^{d}+\overline{z}^{d}), \qquad t >0, \, d \in \mathbb{N}. $$ The associated droplet reveals a topological phase transition: for $t > 1/\sqrt{d}$, it consists of $d$ connected components; whereas for $t < 1/\sqrt{d}$, it becomes simply connected and contains the origin, where a conformal singularity arises. In both regimes, we establish the asymptotic expansion $$ \log Z_n = C_1 n^2 + C_2 n \log n + C_3 n + C_4 \log n + C_5 + O(n^{-1}), $$ as $n \to \infty$, and derive all coefficients explicitly. In the multi-component regime $t > 1/\sqrt{d}$, the constant term $C_5$ exhibits an oscillatory behaviour that depends on the congruence class of $n$ modulo $d$. In particular, in the special case $c = 0$ with $n$ divisible by $d$, our result confirms a conjecture of Deaño and Simm. In contrast, in the conformal singularity regime $t < 1/\sqrt{d}$, the oscillatory behaviour disappears, while additional contributions in $C_4$ arise beyond the scope of the conjecture of Jancovici et al. As a special case $d = 1$, our result yields the asymptotic expansion for the moments of the characteristic polynomial of the complex Ginibre ensemble with finite exponent. In the bulk regime, we further derive the full expansion in powers of $1/N$, thereby providing a precise evaluation of the error term in the result of Webb and Wong.

math-ph↗
arXiv2025

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight

We study $q$-deformed random unitary ensembles associated with the weight function of the Al-Salam--Carlitz orthogonal polynomials, indexed by a parameter $a < 0$. In the special case $a = -1$, the model reduces to the $q$-deformed Gaussian unitary ensemble. Employing the Flajolet--Viennot theory together with the combinatorics of matchings, we derive an explicit positive-sum expression for the spectral moments. In the double-scaling regime $q = e^{-λ/N}$, where $N$ denotes the ensemble size and $λ> 0$ is fixed, we derive the first two terms in the large-$N$ expansion of the spectral moments. As a consequence, we obtain a closed-form expression for the limiting spectral density. Notably, this density exhibits two successive phase transitions as $λ$ increases, characterised by a reduction in the number of soft edges from two, to one, and eventually to none. Furthermore, we show that the limiting density coincides with the limiting zero distribution of the Al-Salam--Carlitz orthogonal polynomials under the same scaling.

math-ph↗
arXiv2025

Orthogonal polynomials in the spherical ensemble with two insertions

We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n, N\rightarrow\infty$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem.

math.CA↗
arXiv2025

On the superintegrability of the Gaussian $β$ ensemble and its $(q,t)$ generalisation

In the present context, superintegrability is a property of certain probability density functions coming from matrix models, which relates to the average over a distinguished basis of symmetric functions, typically the Jack or Macdonald polynomials. It states that the average can be computed according a certain combination of those same polynomials, now specialised by specific substitutions when expressed in terms of the power sum basis. For a particular $(q,t)$ generalisation of the Gaussian $β$ ensemble from random matrix theory, known independently from the consideration of certain integrable gauge theories, we use results developed in a theory of multivariable Al-Salam and Carlitz polynomials based on Macdonald polynomials to prove the superintegrability identity. This then is used to deduce a duality formula for these same averages, which in turn allows for a derivation of a functional equation for the spectral moments.

math-ph↗