Microscopic description of log and Coulomb
                    gases

Serfaty, Sylvia

doi:10.1090/pcms/026/08

Cited by 25 publications

(28 citation statements)

References 39 publications

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“…The limiting spectral density is constant on a disc of radius O( √ N ) for all three Ginibre ensembles, and also for Coulomb gases (4) for all β > 0, see e.g. [49] for a review. The local bulk statistics is defined by zooming into the vicinity of radius R = O(1) of a few mean level spacings around a bulk eigenvalue z 0 , chosen far away from the real axis and the edge of the support.…”

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confidence: 99%

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

et al. 2019

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We study the transition between integrable and chaotic behaviour in dissipative open quantum systems, exemplified by a boundary driven quantum spin-chain. The repulsion between the complex eigenvalues of the corresponding Liouville operator in radial distance s is used as a universal measure. The corresponding level spacing distribution is well fitted by that of a static two-dimensional Coulomb gas with harmonic potential at inverse temperature β ∈ [0, 2]. Here, β = 0 yields the two-dimensional Poisson distribution, matching the integrable limit of the system, and β = 2 equals the distribution obtained from the complex Ginibre ensemble, describing the fully chaotic limit. Our findings generalise the results of Grobe, Haake and Sommers who derived a universal cubic level repulsion for small spacings s. We collect mathematical evidence for the universality of the full level spacing distribution in the fully chaotic limit at β = 2. It holds for all three Ginibre ensembles of random matrices with independent real, complex or quaternion matrix elements. arXiv:1910.03520v4 [cond-mat.stat-mech]

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confidence: 99%

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

et al. 2019

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“…Only at the particular value β = 2 the point process is determinantal, and local universality has been shown for invariant (see e.g., [2,8,12,33,37]) and Wigner ensembles [49]. For general β the low temperature limit corresponding to β 1 is subject of on going research (see e.g., [6]), due to the conjectured condensation on the so-called Abrikosov lattice, and we refer to [48] for a recent review and references.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Notice that the last term on the right-hand side above will also contain the density, but is of sub-leading order. We also refer the reader to [23,42] for a different approach using the integral representation for the delta function in (48). Let us discuss the two different scaling large-N limits (8) and (9) of the free energy (51), starting with the more standard limit (9).…”

Section: The Free Energy In 2d and 1dmentioning

confidence: 99%

The High Temperature Crossover for General 2D Coulomb Gases

Akemann

Byun

2019

J Stat Phys

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We consider N particles in the plane influenced by a general external potential that are subject to the Coulomb interaction in two dimensions at inverse temperature β . At large temperature, when scaling β = 2c/N with some fixed constant c > 0, in the large-N limit we observe a crossover from Ginibre's circular law or its generalization to the density of non-interacting particles at β = 0. Using several different methods we derive a partial differential equation of generalized Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential and present the analytic solution for the density in case of a Gaussian plus logarithmic potential.

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“…The relations (4.3)-(4.7) can be inserted into the Gibbs measure (1.7) to yield so-called "concentration results" in the case with temperature, see [Se3] (for prior such concentration results, see [MMS,BG1,CHM]).…”

Section: Deterministic Dynamics Results -Problems (3) For General Rementioning

confidence: 99%

Systems of Points with Coulomb Interactions

Serfaty

2018

EMS Newsl.

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Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as "the mean-field limit" results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions. MSC : 60F05,

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Microscopic description of log and Coulomb gases

Cited by 25 publications

References 39 publications

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems

The High Temperature Crossover for General 2D Coulomb Gases

Systems of Points with Coulomb Interactions

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