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

Byun, Sung-Soo

doi:10.1007/s40315-023-00494-4

Comput. Methods Funct. Theory

2023

DOI: 10.1007/s40315-023-00494-4

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

Sung-Soo Byun

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2024

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Cited by 3 publications

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Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Ameur,

Charlier,

Cronvall

2024

J Stat Phys

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We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant $$\Gamma =2$$ Γ = 2 and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/n from the hard edge. At distances much larger than $$1/\sqrt{n}$$ 1 / n , the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel $$K_{n}(z,w)$$ K n ( z , w ) as $$n\rightarrow \infty $$ n → ∞ in two microscopic regimes (with either $$|z-w| = \mathcal{O}(1/\sqrt{n})$$ | z - w | = O ( 1 / n ) or $$|z-w| = \mathcal{O}(1/n)$$ | z - w | = O ( 1 / n ) ), as well as in three macroscopic regimes (with $$|z-w| \asymp 1$$ | z - w | ≍ 1 ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.

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Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Ameur,

Charlier,

Cronvall

2024

J Stat Phys

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Linear statistics for Coulomb gases: higher order cumulants

De Bruyne,

Le Doussal,

Majumdar

et al. 2024

J. Phys. A: Math. Theor.

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We consider N classical particles interacting via the Coulomb potential in spatial dimension d and in the presence of an external trap, at equilibrium at inverse temperature β. In the large N limit, the particles are conﬁned within a droplet of ﬁnite size. We study smooth linear statistics, i.e. the ﬂuctuations of sums of the form LN = PNi=1 f(xi), where xi’s are the positions of the particles and where f(xi) is a suﬃciently regular function. There exists at present standard results for the ﬁrst and second moments of LNin the large N limit, as well as associated Central Limit Theorems in general dimension and for a wide class of conﬁning potentials. Here we obtain explicit expressions for the higher order cumulants of LN at large N, when the function f(x) = f(|x|) and the conﬁning potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of f′(|x|) and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a d-dimensional sphere. In the particular two-dimensional case d = 2 at the special value β = 2, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of LN.

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Finite size corrections for real eigenvalues of the elliptic Ginibre matrices

Byun,

Lee

2024

Random Matrices: Theory Appl.

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In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.

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

Cited by 3 publications

References 50 publications

Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall

Linear statistics for Coulomb gases: higher order cumulants

Finite size corrections for real eigenvalues of the elliptic Ginibre matrices

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