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 Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 6, Pages 43–64 (Mi izv8385)

This article is cited in 3 scientific papers (total in 3 papers)

Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. III. The infinite Bessel process as the limit of the radial parts of finite-dimensional projections of infinite Pickrell measures

A. I. Bufetovabcd

a Steklov Mathematical Institute of Russian Academy of Sciences
b Institute for Information Transmission Problems, Russian Academy of Sciences
c National Research University "Higher School of Economics", Moscow
d Aix-Marseille Université, CNRS, Centrale Marseille Institut de Mathématiques de Marseille

Abstract: In the third paper of the series we complete the proof of our main result: a description of the ergodic decomposition of infinite Pickrell measures. We first prove that the scaling limit of the determinantal measures corresponding to the radial parts of Pickrell measures is precisely the infinite Bessel process introduced in the first paper of the series. We prove that the ‘Gaussian parameter’ for ergodic components vanishes almost surely. To do this, we associate a finite measure with each configuration and establish convergence to the scaling limit in the space of finite measures on the space of finite measures. We finally prove that the Pickrell measures corresponding to different values of the parameter are mutually singular.

Keywords: weak convergence, the Harish-Chandra–Itzykson–Zuber integral, infinite Bessel process, Jacobi polynomials.

 Funding Agency Grant Number European Research Council 647133 (ICHAOS) Ministry of Education and Science of the Russian Federation ÌÄ-5991.2016.15-100 This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 647133 (ICHAOS)), grant MD no. 5991.2016.1 of the President of the Russian Federation, and has also been funded by the Russian Academic Excellence Project ‘5-100’.

DOI: https://doi.org/10.4213/im8385

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English version:
Izvestiya: Mathematics, 2016, 80:6, 1035–1056

Bibliographic databases:

UDC: 517.938+519.21
MSC: 22D40, 28C10, 28D15, 43A05, 60B15, 60G55
Received: 07.04.2015
Revised: 16.10.2015

Citation: A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. III. The infinite Bessel process as the limit of the radial parts of finite-dimensional projections of infinite Pickrell measures”, Izv. RAN. Ser. Mat., 80:6 (2016), 43–64; Izv. Math., 80:6 (2016), 1035–1056

Citation in format AMSBIB
\Bibitem{Buf16} \by A.~I.~Bufetov \paper Infinite determinantal measures and the ergodic decomposition of infinite Pickrell~measures.~III. The infinite Bessel process as the limit of the radial parts of finite-dimensional projections of infinite Pickrell measures \jour Izv. RAN. Ser. Mat. \yr 2016 \vol 80 \issue 6 \pages 43--64 \mathnet{http://mi.mathnet.ru/izv8385} \crossref{https://doi.org/10.4213/im8385} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3588812} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2016IzMat..80.1035B} \elib{https://elibrary.ru/item.asp?id=27484921} \transl \jour Izv. Math. \yr 2016 \vol 80 \issue 6 \pages 1035--1056 \crossref{https://doi.org/10.1070/IM8385} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000393621500002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85011708017} 

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This publication is cited in the following articles:
1. A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. II. Convergence of infinite determinantal measures”, Izv. Math., 80:2 (2016), 299–315
2. Y. Qiu, “Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures”, Adv. Math., 308 (2017), 1209–1268
3. Theodoros Assiotis, “Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices”, SIGMA, 15 (2019), 067, 24 pp.
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