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A Convergent \(\varvec{\frac{1}{N}}\) Expansion for GUE

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Abstract

We show that the asymptotic 1 / N expansion for the averages of linear statistics of the GUE is convergent when the test function is an entire function of order two and finite type. This allows to fully recover the mean eigenvalue density function for finite N from the coefficients of the expansion thus providing a resummation procedure. As an intermediate result, we compute the bilateral Laplace transform of the GUE reproducing kernel in the half-sum variable, generalizing a formula of Haagerup and Thorbjørnsen.

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Acknowledgments

I am grateful to my supervisor, Sasha Sodin, for his continuous help and advice. I also would like to thank Yan Fyodorov for helpful discussions and for a suggestion leading to the current version of Theorem 2. I would like to thank Andrei Iacob for copy editing the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Offer Kopelevitch

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  1. Offer Kopelevitch
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Correspondence to Offer Kopelevitch.

Additional information

Communicated by Anton Bovier.

Supported in part by the European Research Council start-up Grant 639305 (SPECTRUM).

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Kopelevitch, O. A Convergent \(\varvec{\frac{1}{N}}\) Expansion for GUE. Ann. Henri Poincaré 19, 3883–3899 (2018). https://doi.org/10.1007/s00023-018-0727-x

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