Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

Akemann, Gernot; Byun, Sung‐Soo; Ebke, Markus

doi:10.1007/s10955-022-03005-2

Cited by 10 publications

(10 citation statements)

References 112 publications

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“…The weight function with a jump-type singularity gives rise to the moment generating function of the disc counting function. It has been extensively studied in recent years [4,8,13,17,20,22]. (We also refer to [26,40,41,51] for physical motivations of these problems from the counting statistics of rotating free fermions.)…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

Byun¹,

Kang²,

Seo³

2022

Preprint

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We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and including the O(1)-terms as the number N of particles increases. Notably, our findings stress that the formulas of the O(log N )-and O(1)-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to a universal additive constant. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the O(N )-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

Byun¹,

Kang²,

Seo³

2022

Preprint

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“…(See also [8] for recent work on the case of the potential W with a hard wall.) In this case, the cumulants are given in terms of poly-logarithms for finite and infinite N. These results were extended to normal symplectic ensembles in the GinSE class for the mean and variance in [1]. It was found that when measuring the variance in units of the mean density given by the Laplacian of the potential ∆W(R) at radius R, the linear growth of the variance in the bulk is universal, i.e.…”

Section: Introductionmentioning

confidence: 92%

“…The expression (2.46) is equivalent to (2.53) due to [52, 3.364.1 and 3.366.1]. The expression (2.49) for β = 4 was shown in [1] as a large-N limit of the number variance (2.94) of the finite symplectic Ginibre ensemble in the origin regime. The second expression (2.50) is new and can be rewritten as Var N (4)…”

Section: Proposition 24 (Number Variance Of the Infinite Ginue And Gi...mentioning

confidence: 99%

“…This physical situation is akin to the well known lowest Landau level problem of electrons in a plane and in the presence of a perpendicular magnetic field [32,45,48]. This connection with the physics of the lowest Landau levels has naturally motivated the study of the FCS in the complex Ginibre ensemble, denoted here as GinUE [1,6,7,14,25,27,28,39,73,90] (for a recent review see [16]), as well as some natural extensions of it, including the higher Landau levels [68,69,91,94], related to the socalled poly-analytic Ginibre ensemble [53]. We also refer to [78,79] and references therein for earlier work on the counting statistics of Hermitian random matrix ensembles and its applications to one-dimensional systems of trapped fermions.…”

Section: Introductionmentioning

confidence: 99%

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

Akemann,

Byun,

Ebke

et al. 2023

J. Phys. A: Math. Theor.

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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.

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“…(See also [18] for an earlier work.) Beyond the gap probabilities, Proposition 5.10 can be used to investigate various counting statistics [33,43,6] as well as fluctuations of the maximal modulus [153,60].…”

Section: Correlation Functions and Sum Rulesmentioning

confidence: 99%

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

Byun¹,

Forrester²

2023

Preprint

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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.

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Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

Cited by 10 publications

References 112 publications

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

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

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

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