Abstract
We study N-particle systems in \(\mathbb {R}^d\) 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 \(N\rightarrow \infty \) for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature \(\beta \). We show that a large deviation principle holds with a rate function of the form ‘\(\beta \)-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 \(\beta \), the limiting density for \(s>d\) is strongly \(\beta \)-dependent.
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Notes
At an April 2018 ICERM workshop, S. Miller announced that, together with H. Cohn, A. Kumar, D. Radchenko and M. Viazovska, the \(E_8\) and Leech lattices are universally optimal, which together with (3.1) verifies the conjecture for \(d = 8\) and \(d = 24\).
That is, \(\overline{P}_{R, M}\in \overline{{\mathcal {M}}}(\Omega \times \mathcal {X}[K_R]).\)
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The authors thank the referee for a careful reading and helpful suggestions.
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Communicated by Peter J. Forrester.
D. P. Hardin and E. B. Saff: The research of these authors was supported, in part, by the U. S. National Science Foundation under the Grant DMS-1516400 and was facilitated by the hospitality and support of the Laboratoire Jacques-Louis Lions at Marie et Pierre Curie Université.
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Hardin, D.P., Leblé, T., Saff, E.B. et al. Large Deviation Principles for Hypersingular Riesz Gases. Constr Approx 48, 61–100 (2018). https://doi.org/10.1007/s00365-018-9431-9
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DOI: https://doi.org/10.1007/s00365-018-9431-9