Large Deviation Principles for Hypersingular Riesz Gases

We study N -particle systems in R d whose interactions are governed by a hypersingular Riesz potential | x - y | - s , s > d , and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as N → ∞ for random point configurations with respect to...

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Bibliographic Details
Published in:Constructive approximation 2018-08, Vol.48 (1), p.61-100
Main Authors: Hardin, Douglas P., Leblé, Thomas, Saff, Edward B., Serfaty, Sylvia
Format: Article
Language:English
Subjects:
Analysis
Asymptotic properties
Constraining
Density
Deviation
Mathematics
Mathematics and Statistics
Numerical Analysis
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Summary:We study N -particle systems in R d whose interactions are governed by a hypersingular Riesz potential | x - y | - s , s > d , and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as N → ∞ for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature β . We show that a large deviation principle holds with a rate function of the form ‘ β -Energy + Entropy’, yielding that the microscopic behavior (on the scale N - 1 / d ) of such N -point systems is asymptotically determined by the minimizers of this rate function. In contrast to the asymptotic behavior in the integrable case s < d , where on the macroscopic scale N -point empirical measures have limiting density independent of β , the limiting density for s > d is strongly β -dependent.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-018-9431-9