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

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Book cover Algebraic and Analytic Microlocal Analysis (AAMA 2013)

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Abstract

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 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.

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Notes

  1. 1.

    General references for this section are the books [33, 60].

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Acknowledgements

It is a pleasure to thank Sébastien Boucksom, David Witt-Nyström, Frédéric Faure and Jeff Steif for stimulating and illuminating discussions. The author is particularly grateful to Bo Berndtsson for helpful discussions concerning Theorem 4.3. Thanks also to the referee for comments that helped to improve the exposition.

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  1. Chalmers University of Technology, Gothenburg, Sweden

    Robert J. Berman

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  1. Robert J. Berman
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Correspondence to Robert J. Berman .

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Editors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, USA

    Michael Hitrik

  2. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Dmitry Tamarkin

  3. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Boris Tsygan

  4. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Steve Zelditch

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Berman, R.J. (2018). Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality. In: Hitrik, M., Tamarkin, D., Tsygan, B., Zelditch, S. (eds) Algebraic and Analytic Microlocal Analysis. AAMA 2013. Springer Proceedings in Mathematics & Statistics, vol 269. Springer, Cham. https://doi.org/10.1007/978-3-030-01588-6_5

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