Exponential moments for disk counting statistics of random normal matrices in the critical regime

Abstract

We obtain large $n$ asymptotics for the $m$-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble. We focus on the critical regime where all disk boundaries are merging at speed $n^{-\frac{1}{2}}$, either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise large $n$ asymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as large $n$ asymptotics for $n\times n$ determinants with merging planar discontinuities.

arXiv comment23 pages, 1 figure

References in corpus (26)

• Characteristic Polynomials of Complex Random Matrix Models
• On the structure of Normal Matrix Model
• Toeplitz determinants with merging singularities
• Induced Ginibre ensemble of random matrices and quantum operations
• Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions
• Entanglement entropy and Berezin-Toeplitz operators
• The random normal matrix model: insertion of a point charge
• Emergence of a singularity for Toeplitz determinants and Painleve V
• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• On the moments of the characteristic polynomial of a Ginibre random matrix
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Gap Probabilities in Non-Hermitian Random Matrix Theory
• Entanglement spectroscopy of chiral edge modes in the Quantum Hall effect
• Large deviations of radial statistics in the two-dimensional one-component plasma
• Fine asymptotic behavior in eigenvalues of random normal matrices: Ellipse Case
• Large gap asymptotics for Airy kernel determinants with discontinuities
• Characteristic polynomials of complex random matrices and Painlevé transcendents
• Discontinuity in the asymptotic behavior of planar orthogonal polynomials under a perturbation of the Gaussian weight
• Counting statistics for non-interacting fermions in a rotating trap
• Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
• Planar Orthogonal Polynomials As Type II Multiple Orthogonal Polynomials
• Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
• On the extreme value statistics of normal random matrices and 2D Coulomb gases: Universality and finite N corrections

Cited by in corpus (5)

• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Functional Central Limit Theorems for Constrained Mittag-Leffler Ensemble in Hard Edge Scaling
• Universality for fluctuations of counting statistics of random normal matrices