# American Institute of Mathematical Sciences

2013, 20: 12-30. doi: 10.3934/era.2013.20.12

## Infinite determinantal measures

 1 Laboratoire d'Analyse, Topologie, Probabilités, Aix-Marseille Université, CNRS, Marseille, France

Received  July 2012 Revised  November 2012 Published  February 2013

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional.
Theorem 4.1, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.
Citation: Alexander I. Bufetov. Infinite determinantal measures. Electronic Research Announcements, 2013, 20: 12-30. doi: 10.3934/era.2013.20.12
##### References:
 [1] V. I. Bogachev, "Measure Theory," Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [2] A. Borodin, Determinantal point processes, in "The Oxford Handbook of Random Matrix Theory," Oxford University Press, Oxford, (2011), 231-249. [3] A. Borodin, A. Okounkov and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc., 13 (2000), 481-515. doi: 10.1090/S0894-0347-00-00337-4. [4] A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures, Comm. Math. Phys., 223 (2001), 87-123. doi: 10.1007/s002200100529. [5] A. Borodin and E. M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys., 121 (2005), 291-317. doi: 10.1007/s10955-005-7583-z. [6] A. I. Bufetov, Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups, arXiv:1105.0664, 2011. [7] A. I. Bufetov, Finiteness of ergodic unitarily invariant measures on spaces of infinite matrices, to appear in Annales de l'Institut Fourier, arXiv:1108.2737, 2011. [8] A. I. Bufetov, Multiplicative functionals of determinantal processes, Uspekhi Mat. Nauk, 67 (2012), 177-178; translation in Russian Math. Surveys, 67 (2012), 181-182. doi: 10.1070/RM2012v067n01ABEH004779. [9] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Determinantal processes and independence, Probab. Surv., 3 (2006), 206-229. doi: 10.1214/154957806000000078. [10] A. Kolmogoroff, "Grundbegriffe der Wahrscheinlichkeitsrechnung," Springer-Verlag, Berlin-New York, 1933. [11] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal., 59 (1975), 219-239. [12] R. Lyons, Determinantal probability measures, Publ. Math. Inst. Hautes Études Sci., 98 (2003), 167-212. doi: 10.1007/s10240-003-0016-0. [13] R. Lyons and J. Steif, Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination, Duke Math. J., 120 (2003), 515-575. doi: 10.1215/S0012-7094-03-12032-3. [14] E. Lytvynov, Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density, Rev. Math. Phys., 14 (2002), 1073-1098. doi: 10.1142/S0129055X02001533. [15] O. Macchi, The coincidence approach to stochastic point processes, Advances in Appl. Probability, 7 (1975), 83-122. [16] Yu. A. Neretin, Hua-type integrals over unitary groups and over projective limits of unitary groups, Duke Math. J., 114 (2002), 239-266. doi: 10.1215/S0012-7094-02-11423-9. [17] G. Olshanski, The quasi-invariance property for the Gamma kernel determinantal measure, Adv. Math., 226 (2011), 2305-2350. doi: 10.1016/j.aim.2010.09.015. [18] G. Olshanski, Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe, in "Representation of Lie Groups and Related Topics," Adv. Stud. Contemp. Math., 7, Gordon and Breach, NY, 1990, 269-463. Available from: http://www.iitp.ru/upload/userpage/52/HoweForm.pdf. [19] G. Olshanski, "Unitary Representations of Infinite-Dimensional Classical Groups," (Russian), D. Sci Thesis, Institute of Geography of the Russian Academy of Sciences, 1989. Available from: http://www.iitp.ru/upload/userpage/52/Olshanski_thesis.pdf. [20] G. Olshanski and A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, in "Contemporary Mathematical Physics," Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, (1996), 137-175. [21] D. Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math., 150 (1991), 139-166. [22] D. Pickrell, Separable representations of automorphism groups of infinite symmetric spaces, J. Funct. Anal., 90 (1990), 1-26. doi: 10.1016/0022-1236(90)90078-Y. [23] D. Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal., 70 (1987), 323-356. doi: 10.1016/0022-1236(87)90116-9. [24] M. Rabaoui, Asymptotic harmonic analysis on the space of square complex matrices, J. Lie Theory, 18 (2008), 645-670. [25] M. Rabaoui, A Bochner type theorem for inductive limits of Gelfand pairs, Ann. Inst. Fourier (Grenoble), 58 (2008), 1551-1573. [26] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. I-IV," Second edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. [27] T. Shirai and Y. Takahashi, Random point fields associated with fermion, boson and other statistics, in "Stochastic Analysis on Large Scale Interacting Systems," Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, (2004), 345-354. [28] T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes, J. Funct. Anal., 205 (2003), 414-463. doi: 10.1016/S0022-1236(03)00171-X. [29] T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their ergodic and Gibbs properties, Ann. Probab., 31 (2003), 1533-1564. doi: 10.1214/aop/1055425789. [30] A. Soshnikov, Determinantal random point fields, (Russian) Uspekhi Mat. Nauk, 55 (2000), 107-160; translation in Russian Math. Surveys, 55 (2000), 923-975. doi: 10.1070/rm2000v055n05ABEH000321. [31] G. Szegö, "Orthogonal Polynomials," AMS, 1969. [32] C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Comm. Math. Phys., 161 (1994), 289-309. [33] A. M. Veršik, A description of invariant measures for actions of certain infinite-dimensional groups, (Russian) Dokl. Akad. Nauk SSSR, 218 (1974), 749-752.

show all references

##### References:
 [1] V. I. Bogachev, "Measure Theory," Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. [2] A. Borodin, Determinantal point processes, in "The Oxford Handbook of Random Matrix Theory," Oxford University Press, Oxford, (2011), 231-249. [3] A. Borodin, A. Okounkov and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc., 13 (2000), 481-515. doi: 10.1090/S0894-0347-00-00337-4. [4] A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures, Comm. Math. Phys., 223 (2001), 87-123. doi: 10.1007/s002200100529. [5] A. Borodin and E. M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys., 121 (2005), 291-317. doi: 10.1007/s10955-005-7583-z. [6] A. I. Bufetov, Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups, arXiv:1105.0664, 2011. [7] A. I. Bufetov, Finiteness of ergodic unitarily invariant measures on spaces of infinite matrices, to appear in Annales de l'Institut Fourier, arXiv:1108.2737, 2011. [8] A. I. Bufetov, Multiplicative functionals of determinantal processes, Uspekhi Mat. Nauk, 67 (2012), 177-178; translation in Russian Math. Surveys, 67 (2012), 181-182. doi: 10.1070/RM2012v067n01ABEH004779. [9] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Determinantal processes and independence, Probab. Surv., 3 (2006), 206-229. doi: 10.1214/154957806000000078. [10] A. Kolmogoroff, "Grundbegriffe der Wahrscheinlichkeitsrechnung," Springer-Verlag, Berlin-New York, 1933. [11] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rational Mech. Anal., 59 (1975), 219-239. [12] R. Lyons, Determinantal probability measures, Publ. Math. Inst. Hautes Études Sci., 98 (2003), 167-212. doi: 10.1007/s10240-003-0016-0. [13] R. Lyons and J. Steif, Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination, Duke Math. J., 120 (2003), 515-575. doi: 10.1215/S0012-7094-03-12032-3. [14] E. Lytvynov, Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density, Rev. Math. Phys., 14 (2002), 1073-1098. doi: 10.1142/S0129055X02001533. [15] O. Macchi, The coincidence approach to stochastic point processes, Advances in Appl. Probability, 7 (1975), 83-122. [16] Yu. A. Neretin, Hua-type integrals over unitary groups and over projective limits of unitary groups, Duke Math. J., 114 (2002), 239-266. doi: 10.1215/S0012-7094-02-11423-9. [17] G. Olshanski, The quasi-invariance property for the Gamma kernel determinantal measure, Adv. Math., 226 (2011), 2305-2350. doi: 10.1016/j.aim.2010.09.015. [18] G. Olshanski, Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe, in "Representation of Lie Groups and Related Topics," Adv. Stud. Contemp. Math., 7, Gordon and Breach, NY, 1990, 269-463. Available from: http://www.iitp.ru/upload/userpage/52/HoweForm.pdf. [19] G. Olshanski, "Unitary Representations of Infinite-Dimensional Classical Groups," (Russian), D. Sci Thesis, Institute of Geography of the Russian Academy of Sciences, 1989. Available from: http://www.iitp.ru/upload/userpage/52/Olshanski_thesis.pdf. [20] G. Olshanski and A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, in "Contemporary Mathematical Physics," Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, (1996), 137-175. [21] D. Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math., 150 (1991), 139-166. [22] D. Pickrell, Separable representations of automorphism groups of infinite symmetric spaces, J. Funct. Anal., 90 (1990), 1-26. doi: 10.1016/0022-1236(90)90078-Y. [23] D. Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal., 70 (1987), 323-356. doi: 10.1016/0022-1236(87)90116-9. [24] M. Rabaoui, Asymptotic harmonic analysis on the space of square complex matrices, J. Lie Theory, 18 (2008), 645-670. [25] M. Rabaoui, A Bochner type theorem for inductive limits of Gelfand pairs, Ann. Inst. Fourier (Grenoble), 58 (2008), 1551-1573. [26] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. I-IV," Second edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. [27] T. Shirai and Y. Takahashi, Random point fields associated with fermion, boson and other statistics, in "Stochastic Analysis on Large Scale Interacting Systems," Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, (2004), 345-354. [28] T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes, J. Funct. Anal., 205 (2003), 414-463. doi: 10.1016/S0022-1236(03)00171-X. [29] T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their ergodic and Gibbs properties, Ann. Probab., 31 (2003), 1533-1564. doi: 10.1214/aop/1055425789. [30] A. Soshnikov, Determinantal random point fields, (Russian) Uspekhi Mat. Nauk, 55 (2000), 107-160; translation in Russian Math. Surveys, 55 (2000), 923-975. doi: 10.1070/rm2000v055n05ABEH000321. [31] G. Szegö, "Orthogonal Polynomials," AMS, 1969. [32] C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Comm. Math. Phys., 161 (1994), 289-309. [33] A. M. Veršik, A description of invariant measures for actions of certain infinite-dimensional groups, (Russian) Dokl. Akad. Nauk SSSR, 218 (1974), 749-752.
 [1] Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks and Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019 [2] Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012 [3] Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006 [4] Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517 [5] Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 [6] Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems and Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567 [7] Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407 [8] Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 [9] Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149 [10] Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83 [11] Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379 [12] Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066 [13] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial and Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [14] Paolo Perfetti. An infinite-dimensional extension of a Poincaré's result concerning the continuation of periodic orbits. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 401-418. doi: 10.3934/dcds.1997.3.401 [15] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [16] Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial and Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503 [17] Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072 [18] Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks and Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267 [19] Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303 [20] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731

2020 Impact Factor: 0.929