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Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry

Volume 123 / 2019

Robert J. Berman Annales Polonici Mathematici 123 (2019), 71-153 MSC: 32U35, 32W20, 60G55, 60F10. DOI: 10.4064/ap180925-4-7 Published online: 23 October 2019

Abstract

This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kähler–Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in $\mathbb C ^{n}.$ A fair amount of background material is included. Along the way the results are generalized to the non-compact setting of $\mathbb C ^{n}.$ This yields a probabilistic construction of Kähler solutions to Einstein’s equations in $\mathbb C ^{n}$, with cosmological constant $-\beta $, from a gas of interpolation nodes in equilibrium at positive inverse temperature $\beta .$ In the infinite temperature limit, $\beta \rightarrow 0$, solutions to the Calabi–Yau equation are obtained. In the opposite zero temperature case the results may be interpreted as “transcendental” analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature $\beta $ playing the role of the degree of a polynomial.

Authors

  • Robert J. BermanMathematical Sciences
    Chalmers University of Technology and the University of Gothenburg
    SE-412 96 Gøteborg, Sweden
    e-mail

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