Wronskian structures of planar symplectic ensembles

Byun, Sung‐Soo; Ebke, Markus; Seo, Seong-Mi

doi:10.1088/1361-6544/aca3f4

Cited by 6 publications

(8 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2.8], the subleading correction term was derived. In the weakly non-Hermitian regime when τ = 1 − α 2 /N, the bulk and edge scaling limits were obtained in [36], extending previous results [118,16]; see also [62]. In particular, it was shown in [36] that the limiting pre-kernel in the bulk scaling limit is of the unified Wronskian form (5.37) with (5.49) f z (u) := 1 2π…”

Section: Correlation Functions and Sum Rulessupporting

confidence: 78%

See 1 more Smart Citation

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Byun¹,

Forrester²

2023

Preprint

View full text Add to dashboard Cite

This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as 2 × 2 complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.

show abstract

Section: Correlation Functions and Sum Rulessupporting

confidence: 78%

“…A structural feature, highlighted in [7,36], is that the kernels (5.31) and (5.34) can be expressed in a unified way (5.37)…”

Section: Correlation Functions and Sum Rulesmentioning

confidence: 99%

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Byun¹,

Forrester²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…These scaling limits have then been extended to the entire real bulk/edge in Ref. 28. Similar problems were studied in Ref.…”

Section: Introduction and Main Resultsmentioning

confidence: 85%

“…29 for the truncated symplectic ensemble and in Ref. 28 for the Ginibre ensemble with boundary confinements. We also refer to Refs.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On the almost‐circular symplectic induced Ginibre ensemble

Byun

Charlier

2022

Stud Appl Math

View full text Add to dashboard Cite

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 in a thin annulus scriptSN$\mathcal {S}_{N}$ of width O()1N$O\left(\frac{1}{N}\right)$ as N→∞$N \rightarrow \infty$. Our main results are the bulk scaling limits of all correlation functions near the real axis, and also away from the real axis. Near the real axis, the limiting correlation functions are Pfaffians with a new correlation kernel, which interpolates the limiting kernels in the bulk of the symplectic Ginibre ensemble and of the antisymmetric Gaussian Hermitian ensemble of odd size. Away from the real axis, the limiting correlation functions are determinants, and the kernel is the same as the one appearing in the bulk limit of almost‐Hermitian random matrices. Furthermore, we obtain precise large N asymptotics for the probability that no points lie outside scriptSN$\mathcal {S}_{N}$, as well as of several other “semi‐large” gap probabilities.

show abstract

Planar Equilibrium Measure Problem in the Quadratic Fields with a Point Charge

Byun

2023

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

Wronskian structures of planar symplectic ensembles

Cited by 6 publications

References 51 publications

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

On the almost‐circular symplectic induced Ginibre ensemble

Planar Equilibrium Measure Problem in the Quadratic Fields with a Point Charge

Contact Info

Product

Resources

About