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June 16, 2022 15:40–16:15, Operators, Functions, Systems: Classical and Modern. Conference in honor of Nikolai Nikolski. Banach Center, Będlewo, Poland
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Normal approximation,
the Gaussian multiplicative chaos,
and excess one for the sine-process
A. I. Bufetovab a Aix-Marseille Université
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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Number of views: |
This page: | 214 | Video files: | 5 |
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Abstract:
The Soshnikov Central Limit Theorem states that scaled additive statistics of the sineprocess converge to the normal law. The first main result of this talk gives a detailed
comparison between the law of an additive, sufficiently Sobolev regular, statistic under
the sine-process and the normal law. The comparison for low frequencies is obtained
by taking the scaling limit in the Borodin-Okounkov-Geronimo-Case formula. The exponential decay for the high frequencies is obtained, under an additional assumption of
holomorphicity in a horizontal strip, with the use of an analogue of the Johansson change
of variable formula; quasi-invariance of the sine-process under compactly supported diffeomorphisms plays a key rˆole in the proof. The corollaries of the normal approximation
theorem include the convergence of the random entire function, the infinite product with
zeros at the particles, to Gaussian multiplicative chaos. A complementary estimate to
the Ghosh completeness theorem follows in turn: indeed, Ghosh proved that reproducing sine-kernels along almost every configuration of the sine-process form a complete set;
it is proved in the talk that if one particle is removed, then the set is still complete;
whereas if two particles are removed from the configuration, then the resulting set is
the zero set for the Paley-Wiener space. The talk extends the results of the preprint
https://arxiv.org/abs/1912.13454
Language: English
Website:
https://sites.google.com/view/operatorsfunctionssystemsclass/home
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