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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 6, Pages 18–64 (Mi izv8383)  

This article is cited in 5 scientific papers (total in 5 papers)

Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures

A. I. Bufetovabcd

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
c National Research University "Higher School of Economics" (HSE), Moscow
d Aix-Marseille Université

Abstract: This paper is the first in a series of three. We give an explicit description of the ergodic decomposition of infinite Pickrell measures on the space of infinite complex matrices. A key role is played by the construction of $\sigma$-finite analogues of determinantal measures on spaces of configurations, including the infinite Bessel process, a scaling limit of the $\sigma$-finite analogues of the Jacobi orthogonal polynomial ensembles. Our main result identifies the infinite Bessel process with the decomposing measure of an infinite Pickrell measure.

Keywords: determinantal processes, infinite determinantal measures, ergodic decomposition, infinite harmonic analysis, infinite unitary group, scaling limits, Jacobi polynomials, Harish-Chandra–Itzykson–Zuber orbit integral.

Funding Agency Grant Number
Agence Nationale de la Recherche ANR-11-IDEX-0001-02
Ministry of Education and Science of the Russian Federation МД-2859.2014.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations 0014-2015-0006
Russian Foundation for Basic Research 13-01-12449 офи_м
This work was supported by the project A*MIDEX (no. ANR-11-IDEX-0001-02) of the French Republic Government Programme ‘Investing in the Future’ carried out by the French National Agency of Scientific Research (ANR). It was also supported by the Programme of Governmental Support of Scientific Research of Young Russian Scholars, Candidates and Doctors of Sciences (grant no. MD-2859.2014.1), the Programme of Fundamental Research of RAS no. I.28P ‘Mathematical problems of modern control theory’ (project no. 0014-2015-0006 ‘Ergodic theory and dynamical systems’), the subsidy for governmental support of leading universities of the Russian Federation aimed at raising their competitiveness among world leading scientific and educational centres, distributed to the National Research University ‘Higher School of Economics’, and the RFBR (grant no. 13-01-12449-ofi_m).


DOI: https://doi.org/10.4213/im8383

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English version:
Izvestiya: Mathematics, 2015, 79:6, 1111–1156

Bibliographic databases:

Document Type: Article
UDC: 517.938+519.21
MSC: 20C32, 22D40, 28C10, 28D15, 43A05, 60B15, 60G55
Received: 06.04.2015

Citation: A. I. Bufetov, “Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures”, Izv. RAN. Ser. Mat., 79:6 (2015), 18–64; Izv. Math., 79:6 (2015), 1111–1156

Citation in format AMSBIB
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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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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. Math., 80:6 (2016), 1035–1056  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. A. I. Bufetov, Y. Qiu, “The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures”, Ark. Mat., 54:2 (2016), 403–435  crossref  mathscinet  zmath  isi  scopus
    4. Alexander I. Bufetov, “A Palm hierarchy for determinantal point processes with the Bessel kernel”, Proc. Steklov Inst. Math., 297 (2017), 90–97  mathnet  crossref  crossref  mathscinet  isi  elib
    5. Y. Qiu, “Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures”, Adv. Math., 308 (2017), 1209–1268  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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