# American Institute of Mathematical Sciences

January  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, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar [2] A. Borodin, Determinantal point processes, in "The Oxford Handbook of Random Matrix Theory,", Oxford University Press, (2011), 231. Google Scholar [3] A. Borodin, A. Okounkov and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups,, J. Amer. Math. Soc., 13 (2000), 481. doi: 10.1090/S0894-0347-00-00337-4. Google Scholar [4] A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures,, Comm. Math. Phys., 223 (2001), 87. doi: 10.1007/s002200100529. Google Scholar [5] A. Borodin and E. M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs,, J. Stat. Phys., 121 (2005), 291. doi: 10.1007/s10955-005-7583-z. Google Scholar [6] A. I. Bufetov, Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups,, \arXiv{1105.0664}, (2011). Google Scholar [7] A. I. Bufetov, Finiteness of ergodic unitarily invariant measures on spaces of infinite matrices,, to appear in Annales de l'Institut Fourier, (2011). Google Scholar [8] A. I. Bufetov, Multiplicative functionals of determinantal processes,, Uspekhi Mat. Nauk, 67 (2012), 177. doi: 10.1070/RM2012v067n01ABEH004779. Google Scholar [9] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Determinantal processes and independence,, Probab. Surv., 3 (2006), 206. doi: 10.1214/154957806000000078. Google Scholar [10] A. Kolmogoroff, "Grundbegriffe der Wahrscheinlichkeitsrechnung,", Springer-Verlag, (1933). Google Scholar [11] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I,, Arch. Rational Mech. Anal., 59 (1975), 219. Google Scholar [12] R. Lyons, Determinantal probability measures,, Publ. Math. Inst. Hautes Études Sci., 98 (2003), 167. doi: 10.1007/s10240-003-0016-0. Google Scholar [13] R. Lyons and J. Steif, Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination,, Duke Math. J., 120 (2003), 515. doi: 10.1215/S0012-7094-03-12032-3. Google Scholar [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. doi: 10.1142/S0129055X02001533. Google Scholar [15] O. Macchi, The coincidence approach to stochastic point processes,, Advances in Appl. Probability, 7 (1975), 83. Google Scholar [16] Yu. A. Neretin, Hua-type integrals over unitary groups and over projective limits of unitary groups,, Duke Math. J., 114 (2002), 239. doi: 10.1215/S0012-7094-02-11423-9. Google Scholar [17] G. Olshanski, The quasi-invariance property for the Gamma kernel determinantal measure,, Adv. Math., 226 (2011), 2305. doi: 10.1016/j.aim.2010.09.015. Google Scholar [18] G. Olshanski, Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe,, in, 7 (1990), 269. Google Scholar [19] G. Olshanski, "Unitary Representations of Infinite-Dimensional Classical Groups,", (Russian), (1989). Google Scholar [20] G. Olshanski and A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices,, in, 175 (1996), 137. Google Scholar [21] D. Pickrell, Mackey analysis of infinite classical motion groups,, Pacific J. Math., 150 (1991), 139. Google Scholar [22] D. Pickrell, Separable representations of automorphism groups of infinite symmetric spaces,, J. Funct. Anal., 90 (1990), 1. doi: 10.1016/0022-1236(90)90078-Y. Google Scholar [23] D. Pickrell, Measures on infinite-dimensional Grassmann manifolds,, J. Funct. Anal., 70 (1987), 323. doi: 10.1016/0022-1236(87)90116-9. Google Scholar [24] M. Rabaoui, Asymptotic harmonic analysis on the space of square complex matrices,, J. Lie Theory, 18 (2008), 645. Google Scholar [25] M. Rabaoui, A Bochner type theorem for inductive limits of Gelfand pairs,, Ann. Inst. Fourier (Grenoble), 58 (2008), 1551. Google Scholar [26] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. I-IV,", Second edition, (1980). Google Scholar [27] T. Shirai and Y. Takahashi, Random point fields associated with fermion, boson and other statistics,, in, 39 (2004), 345. Google Scholar [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. doi: 10.1016/S0022-1236(03)00171-X. Google Scholar [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. doi: 10.1214/aop/1055425789. Google Scholar [30] A. Soshnikov, Determinantal random point fields,, (Russian) Uspekhi Mat. Nauk, 55 (2000), 107. doi: 10.1070/rm2000v055n05ABEH000321. Google Scholar [31] G. Szegö, "Orthogonal Polynomials,", AMS, (1969). Google Scholar [32] C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel,, Comm. Math. Phys., 161 (1994), 289. Google Scholar [33] A. M. Veršik, A description of invariant measures for actions of certain infinite-dimensional groups,, (Russian) Dokl. Akad. Nauk SSSR, 218 (1974), 749. Google Scholar

show all references

##### References:
 [1] V. I. Bogachev, "Measure Theory,", Vol. II, (2007). doi: 10.1007/978-3-540-34514-5. Google Scholar [2] A. Borodin, Determinantal point processes, in "The Oxford Handbook of Random Matrix Theory,", Oxford University Press, (2011), 231. Google Scholar [3] A. Borodin, A. Okounkov and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups,, J. Amer. Math. Soc., 13 (2000), 481. doi: 10.1090/S0894-0347-00-00337-4. Google Scholar [4] A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures,, Comm. Math. Phys., 223 (2001), 87. doi: 10.1007/s002200100529. Google Scholar [5] A. Borodin and E. M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs,, J. Stat. Phys., 121 (2005), 291. doi: 10.1007/s10955-005-7583-z. Google Scholar [6] A. I. Bufetov, Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups,, \arXiv{1105.0664}, (2011). Google Scholar [7] A. I. Bufetov, Finiteness of ergodic unitarily invariant measures on spaces of infinite matrices,, to appear in Annales de l'Institut Fourier, (2011). Google Scholar [8] A. I. Bufetov, Multiplicative functionals of determinantal processes,, Uspekhi Mat. Nauk, 67 (2012), 177. doi: 10.1070/RM2012v067n01ABEH004779. Google Scholar [9] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Determinantal processes and independence,, Probab. Surv., 3 (2006), 206. doi: 10.1214/154957806000000078. Google Scholar [10] A. Kolmogoroff, "Grundbegriffe der Wahrscheinlichkeitsrechnung,", Springer-Verlag, (1933). Google Scholar [11] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I,, Arch. Rational Mech. Anal., 59 (1975), 219. Google Scholar [12] R. Lyons, Determinantal probability measures,, Publ. Math. Inst. Hautes Études Sci., 98 (2003), 167. doi: 10.1007/s10240-003-0016-0. Google Scholar [13] R. Lyons and J. Steif, Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination,, Duke Math. J., 120 (2003), 515. doi: 10.1215/S0012-7094-03-12032-3. Google Scholar [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. doi: 10.1142/S0129055X02001533. Google Scholar [15] O. Macchi, The coincidence approach to stochastic point processes,, Advances in Appl. Probability, 7 (1975), 83. Google Scholar [16] Yu. A. Neretin, Hua-type integrals over unitary groups and over projective limits of unitary groups,, Duke Math. J., 114 (2002), 239. doi: 10.1215/S0012-7094-02-11423-9. Google Scholar [17] G. Olshanski, The quasi-invariance property for the Gamma kernel determinantal measure,, Adv. Math., 226 (2011), 2305. doi: 10.1016/j.aim.2010.09.015. Google Scholar [18] G. Olshanski, Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe,, in, 7 (1990), 269. Google Scholar [19] G. Olshanski, "Unitary Representations of Infinite-Dimensional Classical Groups,", (Russian), (1989). Google Scholar [20] G. Olshanski and A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices,, in, 175 (1996), 137. Google Scholar [21] D. Pickrell, Mackey analysis of infinite classical motion groups,, Pacific J. Math., 150 (1991), 139. Google Scholar [22] D. Pickrell, Separable representations of automorphism groups of infinite symmetric spaces,, J. Funct. Anal., 90 (1990), 1. doi: 10.1016/0022-1236(90)90078-Y. Google Scholar [23] D. Pickrell, Measures on infinite-dimensional Grassmann manifolds,, J. Funct. Anal., 70 (1987), 323. doi: 10.1016/0022-1236(87)90116-9. Google Scholar [24] M. Rabaoui, Asymptotic harmonic analysis on the space of square complex matrices,, J. Lie Theory, 18 (2008), 645. Google Scholar [25] M. Rabaoui, A Bochner type theorem for inductive limits of Gelfand pairs,, Ann. Inst. Fourier (Grenoble), 58 (2008), 1551. Google Scholar [26] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. Vol. I-IV,", Second edition, (1980). Google Scholar [27] T. Shirai and Y. Takahashi, Random point fields associated with fermion, boson and other statistics,, in, 39 (2004), 345. Google Scholar [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. doi: 10.1016/S0022-1236(03)00171-X. Google Scholar [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. doi: 10.1214/aop/1055425789. Google Scholar [30] A. Soshnikov, Determinantal random point fields,, (Russian) Uspekhi Mat. Nauk, 55 (2000), 107. doi: 10.1070/rm2000v055n05ABEH000321. Google Scholar [31] G. Szegö, "Orthogonal Polynomials,", AMS, (1969). Google Scholar [32] C. A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel,, Comm. Math. Phys., 161 (1994), 289. Google Scholar [33] A. M. Veršik, A description of invariant measures for actions of certain infinite-dimensional groups,, (Russian) Dokl. Akad. Nauk SSSR, 218 (1974), 749. Google Scholar
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