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A survey on determinantal point processes. (English) Zbl 1487.60094

Inahama, Yuzuru (ed.) et al., Stochastic analysis, random fields and integrable probability – Fukuoka 2019. Proceedings of the 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI), Kyushu University, Japan, 31 July – 9 August 2019. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 87, 59-89 (2021).
Summary: We present a first introduction to determinantal point processes, focussing on multiplicative functionals, Palm theory, and key applications such as random matrices, Young diagrams and random holomorphic functions.
For the entire collection see [Zbl 1482.60003].

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
32A36 Bergman spaces of functions in several complex variables
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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[1] Alexei Borodin, Andrei Okounkov and Grigori Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc., 13(3): 481-515, 2000. · Zbl 0938.05061
[2] Alexander I. Bufetov, On multiplicative functionals of determinantal pro-cesses, Uspekhi Mat. Nauk, 67(1(403)):177-178, 2012. · Zbl 1252.60047
[3] Alexander I. Bufetov, On the Vershik-Kerov conjecture concerning the Shannon-McMillan-Breiman theorem for the Plancherel family of mea-sures on the space of Young diagrams, Geom. Funct. Anal., 22(4):938-975, 2012. · Zbl 1254.05024
[4] Alexander I. Bufetov, Infinite determinantal measures, Electron. Res. An-nounc. Math. Sci., 20:12-30, 2013. · Zbl 1287.60064
[5] Alexander I. Bufetov, Quasi-symmetries of determinantal point processes, Ann. Probab., 46(2):956-1003, 2018. · Zbl 1430.60045
[6] Alexander I. Bufetov, The sine-process has excess one, arXiv e-prints, arXiv:1912.13454, December 2019.
[7] Alexander I. Bufetov, Yoann Dabrowski and Yanqi Qiu, Linear rigidity of stationary stochastic processes, Ergodic Theory Dynam. Systems, 38(7): 2493-2507, 2018. · Zbl 1400.37010
[8] Alexander I. Bufetov and Andrey V. Dymov, A functional limit theorem for the sine-process, Int. Math. Res. Not. IMRN, (1):249-319, 2019. · Zbl 1479.60068
[9] Alexander I. Bufetov, Shilei Fan and Yanqi Qiu, Equivalence of Palm mea-sures for determinantal point processes governed by Bergman kernels, Probab. Theory Related Fields, 172(1-2):31-69, 2018. · Zbl 1429.60047
[10] Alexander I. Bufetov and Yanqi Qiu, Determinantal point processes associ-ated with Hilbert spaces of holomorphic functions, Comm. Math. Phys., 351(1):1-44, 2017. · Zbl 1406.60073
[11] Alexander I. Bufetov, Yanqi Qiu and Alexander Shamov, Kernels of condi-tional determinantal measures and the proof of the Lyons-Peres Conjec-ture, arXiv e-prints, to appear in J. Eur. Math. Soc., arXiv:1612.06751, December 2016.
[12] Alexander I. Bufetov and Roman V. Romanov, Division subspaces and integrable kernels, Bull. Lond. Math. Soc., 51(2):267-277, 2019. · Zbl 07094880
[13] Michael Christ, On the ∂ equation in weighted L 2 norms in C 1 , J. Geom. Anal., 1(3):193-230, 1991. · Zbl 0737.35011
[14] D. J. Daley and D. Vere-Jones, An introduction to the theory of point pro-cesses. Vol. I, Probability and its Applications (New York), Springer-Verlag, New York, second edition, 2003, Elementary theory and methods. · Zbl 1026.60061
[15] D. J. Daley and D. Vere-Jones, An introduction to the theory of point pro-cesses. Vol. II, Probability and its Applications (New York), Springer, New York, second edition, 2008, General theory and structure. · Zbl 0657.60069
[16] Henrik Delin, Pointwise estimates for the weighted Bergman projection ker-nel in C n , using a weighted L 2 estimate for the ∂ equation, Ann. Inst. Fourier (Grenoble), 48(4):967-997, 1998. · Zbl 0918.32007
[17] Subhroshekhar Ghosh, Determinantal processes and completeness of ran-dom exponentials: the critical case, Probab. Theory Related Fields, 163(3-4):643-665, 2015. · Zbl 1334.60083
[18] Subhroshekhar Ghosh and Yuval Peres, Rigidity and tolerance in point pro-cesses: Gaussian zeros and Ginibre eigenvalues, Duke Math. J., 166(10): 1789-1858, 2017. · Zbl 1405.60067
[19] Haakan Hedenmalm, Boris Korenblum and Kehe Zhu, Theory of Bergman spaces, volume 199 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. · Zbl 0955.32003
[20] John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. · Zbl 0135.22701
[21] Alexander E. Holroyd and Terry Soo, Insertion and deletion tolerance of point processes, Electron. J. Probab., 18:no. 74, 24, 2013. · Zbl 1291.60101
[22] Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2), 153(1):259-296, 2001. · Zbl 0984.15020
[23] Olav Kallenberg, Random measures, Akademie-Verlag, Berlin; Academic Press, Inc., London, fourth edition, 1986. · Zbl 0345.60032
[24] A. Lenard, Correlation functions and the uniqueness of the state in classical statistical mechanics, Comm. Math. Phys., 30:35-44, 1973.
[25] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal., 59(3):219-239, 1975.
[26] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Arch. Ra-tional Mech. Anal., 59(3):241-256, 1975.
[27] B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math., 26(2):206-222, 1977. · Zbl 0363.62068
[28] Russell Lyons, Determinantal probability measures, Publ. Math. Inst. HautesÉtudes Sci., (98):167-212, 2003. · Zbl 1055.60003
[29] Russell Lyons, Determinantal probability: basic properties and conjectures, In: Proceedings of the International Congress of Mathematicians-Seoul 2014. Vol. IV, pages 137-161, Kyung Moon Sa, Seoul, 2014. · Zbl 1373.60087
[30] Russell Lyons and Jeffrey E. Steif, Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination, Duke Math. J., 120(3):515-575, 2003. · Zbl 1068.82010
[31] Odile Macchi, The coincidence approach to stochastic point processes, Ad-vances in Appl. Probability, 7:83-122, 1975. · Zbl 0366.60081
[32] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, volume 93 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2002. Model operators and sys-tems, Translated from the French by Andreas Hartmann and revised by the author. · Zbl 1007.47002
[33] Yuval Peres and Bálint Virág, Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Math., 194(1):1-35, 2005. · Zbl 1099.60037
[34] G. Baverez, A. I. Bufetov and Y. Qiu
[35] Alexander P. Schuster and Dror Varolin, Toeplitz operators and Carleson measures on generalized Bargmann-Fock spaces, Integral Equations Op-erator Theory, 72(3):363-392, 2012. · Zbl 1262.47047
[36] Kristian Seip, On Korenblum’s density condition for the zero sequences of A −α , J. Anal. Math., 67:307-322, 1995. · Zbl 0845.30014
[37] Tomoyuki Shirai and Yoichiro Takahashi, Fermion process and Fredholm determinant, In: Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999), volume 7 of Int. Soc. Anal. Appl. Comput., pages 15-23, Kluwer Acad. Publ., Dordrecht, 2000. · Zbl 1036.60045
[38] Tomoyuki Shirai and Yoichiro Takahashi, Random point fields associated with certain Fredholm determinants. I: Fermion, Poisson and boson point processes, J. Funct. Anal., 205(2):414-463, 2003. · Zbl 1051.60052
[39] Barry Simon, Trace ideals and their applications, volume 35 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge-New York, 1979. · Zbl 0423.47001
[40] A. Soshnikov, Determinantal random point fields, Uspekhi Mat. Nauk, 55(5(335)):107-160, 2000. · Zbl 0991.60038
[41] A. M. Veršik and S. V. Kerov, Asymptotic behavior of the Plancherel mea-sure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR, 233(6):1024-1027, 1977 · Zbl 0406.05008
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