Rescaling Ward identities in the random normal matrix model

Ameur, Yacin; Kang, Namgyu; Makarov, Nikolai

doi:10.48550/arxiv.1410.4132

Cited by 4 publications

(10 citation statements)

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“…Very recently, the results in [4] concerning X = C have been generalized to less regular data φ [55,7] (with u assumed almost C 4 −smooth; see Section 7.2). As for the scaling limits of the correlation function at the boundary/edge of the support they were established in [5] under suitable regularity and symmetry assumptions. It would be very interesting to consider the behavior at the boundary in higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

Berman

2018

Algebraic and Analytic Microlocal Analysis

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We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Ampère operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials. 7 9 42

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Section: Introductionmentioning

confidence: 99%

Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

Berman

2018

Algebraic and Analytic Microlocal Analysis

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“…Furthermore, yet another limiting kernel exists, which is obtained when imposing a hard edge (at the otherwise soft edge) for the Ginibre ensemble, cf. [6,Theorem 2.3]. Apparently the role of a hard edge differs when imposed for a confining potential as for Ginibre, or for a non-confining potential as here, generalising the Jacobi ensemble.…”

Section: Weak Non-hermiticity In the Bulkmentioning

confidence: 99%

“…The case of a growing droplet where particles are constantly fed in has applications to viscous fluids, or more generally can be viewed as Laplacian growth models [5]. The case of a hard wall imposed at the edge of the droplet has been studied in [6]. When forcing the gas away from its equilibrium position, phase transitions may occur, see [7] for more general potentials and the general situation in d dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

Nagao

Akemann

Kieburg

et al. 2020

J. Phys. A: Math. Theor.

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We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature β = 2 we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all k-point correlation functions for a fixed number of particles N . These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-N limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine-and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.

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“…In another class of related models, which describes, e.g., a one-component plasma (2dimensional Coulomb gas), the charges are related to the eigenvalues of random normal matrices. Hence, the questions of existence and universality of scaling limits for the eigenvalues of a random normal matrix are in focus of these studies (see, e.g., [1], [2] and [3], and the reference therein). Note, that these planar models (including the one-dimensional case) treat only logarithmic interactions.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in the one-dimensional Coulomb gas ensembles

Turova

2018

Ann. Appl. Probab.

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We consider the system of particles on a finite interval with pair-wise nearest neighbours interaction and external force. This model was introduced by Malyshev to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states) with external force. We derive the asymptotic for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity.

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Rescaling Ward identities in the random normal matrix model

Cited by 4 publications

References 0 publications

Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

Phase transitions in the one-dimensional Coulomb gas ensembles

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