A review of exact results for fluctuation formulas in random matrix theory

Forrester, Peter J.

doi:10.1214/23-ps15

Cited by 6 publications

(1 citation statement)

References 151 publications

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“…The large N functional form (B.3) for the characteristic function of B implies that the centred linear statistic B − μN,B is a mean zero Gaussian with variance given by (B.4). The structure of the latter is familiar from the study of the fluctuations associated with a linear statistic for the complex Ginibre ensemble; see the recent review [35,Section 3.5].…”

Section: B Appendixmentioning

confidence: 99%

Spherical Induced Ensembles with Symplectic Symmetry

Byun¹,

Forrester²

2023

SIGMA

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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.

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Section: B Appendixmentioning

confidence: 99%

Spherical Induced Ensembles with Symplectic Symmetry

Byun¹,

Forrester²

2023

SIGMA

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Power spectra of Dyson’s circular ensembles

Forrester,

Witte

2025

Physica D: Nonlinear Phenomena

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Coulomb and Riesz gases: The known and the unknown

Lewin

2022

Journal of Mathematical Physics

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We review what is known, unknown, and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in [Formula: see text] interacting with the Riesz potential ±| x|− s (respectively, −log | x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g., in random matrix theory). The main question addressed in this Review is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader, we give the detail of what is known in the short range case s > d. Finally, we discuss phase transitions and mention what is expected on physical grounds.

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A review of exact results for fluctuation formulas in random matrix theory

Cited by 6 publications

References 151 publications

Spherical Induced Ensembles with Symplectic Symmetry

Spherical Induced Ensembles with Symplectic Symmetry

Power spectra of Dyson’s circular ensembles

Coulomb and Riesz gases: The known and the unknown

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