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 Tr. Mat. Inst. Steklova, 2018, Volume 301, Pages 182–191 (Mi tm3912)

On expanding neighborhoods of local universality of Gaussian unitary ensembles

M. A. Lapik, D. N. Tulyakov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

Abstract: The classical universality theorem states that the Christoffel–Darboux kernel of the Hermite polynomials scaled by a factor of $1/\sqrt n$ tends to the sine kernel in local variables $\widetilde x,\widetilde y$ in a neighborhood of a point $x^*\in(-\sqrt2,\sqrt2)$. This classical result is well known for $\widetilde x,\widetilde y\in K\Subset\mathbb R$. In this paper, we show that this classical result remains valid for expanding compact sets $K=K(n)$. An interesting phenomenon of admissible dependence of the expansion rate of compact sets $K(n)$ on $x^*$ is established. For $x^*\in(-\sqrt2,\sqrt2)\setminus\{0\}$ and for $x^*=0$, there are different growth regimes of compact sets $K(n)$. A transient regime is found.

 Funding Agency Grant Number Russian Science Foundation 14-21-00025 This work is supported by the Russian Science Foundation under grant 14-21-00025.

DOI: https://doi.org/10.1134/S0371968518020139

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 301, 170–179

Bibliographic databases:

Document Type: Article
UDC: 517.53

Citation: M. A. Lapik, D. N. Tulyakov, “On expanding neighborhoods of local universality of Gaussian unitary ensembles”, Complex analysis, mathematical physics, and applications, Collected papers, Tr. Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 182–191; Proc. Steklov Inst. Math., 301 (2018), 170–179

Citation in format AMSBIB
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