Abstract
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space \(A^2(U, \omega )\) over a domain \(U \subset \mathbb {C}^d\), we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain U contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the \(H^\infty (U)\)-module structure of \(A^2(U, \omega )\). A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of U.
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References
Balk, M.B.: Polyanalytic functions and their generalizations. In: Complex Analysis, I. Encyclopaedia Mathematical Science, vol. 85, pp. 195–253. Springer, Berlin (1997)
Bergman, S.: The Kernel Function and Conformal Mapping, vol. 5. American Mathematical Society, Providence RI (1970)
Bufetov, A.I.: Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures. Izv. Ross. Akad. Nauk Ser. Mat. 79(6), 18–64 (2015)
Bufetov, A.I.: Quasi-symmetries of determinantal point processes. arXiv:1409.2068
Bufetov, A.I., Qiu, Y.: Determinantal point processes associated with Hilbert spaces of holomorphic functions. Commun. Math. Phys. 351(1), 1–44 (2017)
Bufetov, A.I., Qiu, Y., Shamov, A.: Kernels of conditional determinantal measures. arXiv:1612.06751
Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes, Vol. 1. In: Probability and its Applications (New York), 2nd edn. Springer, New York (2003). Elementary theory and methods
Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Relat. Fields, pp. 1–23 (2014)
Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. To appear in Duke Math. J. arXiv:1211.3506
Haimi, A., Hedenmalm, H.: The polyanalytic Ginibre ensembles. J. Stat. Phys. 153(1), 10–47 (2013)
Haimi, A., Hedenmalm, H.: Asymptotic expansion of polyanalytic Bergman kernels. J. Funct. Anal. 267(12), 4667–4731 (2014)
Holroyd, A.E., Soo, T.: Insertion and deletion tolerance of point processes. Electron. J. Probab. 18(74), 24 (2013)
Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)
Kallenberg, O.: Random Measures, 4th edn. Akademie-Verlag, Berlin (1986)
Khintchine, A.Ya.: Mathematical methods of queuing theory. Proceedings of the Steklov Institute 49, 3–122 (1955)
Krantz, S.G.: Geometric Analysis of the Bergman Kernel and Metric, Graduate Texts in Mathematics, vol. 268. Springer, New York (2013)
Lindvall, T.: On Strassen’s theorem on stochastic domination. Electron. Commun. Probab. 4, 51–59 (1999)
Lyons, R.: Determinantal probability: basic properties and conjectures. In: Proceedings of the International Congress of Mathematicians 2014, vol. IV, pp. 137–161. Seoul (2014)
Macchi, O.: The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7, 83–122 (1975)
Osada, H., Shirai, T.: Absolute continuity and singularity of Palm measures of the Ginibre point process. Probab. Theory Relat. Fields 165(3–4), 725–770 (2016)
Palm, C.: Intensitätsschwankungen im Fernsprechverkehr. Ericsson Tech. 44, 1–189 (1943)
Peres, Y., Virág, B.: Zeros of the iid Gaussian power series: a conformally invariant determinantal process. Acta Math. 194(1), 1–35 (2005)
Qiu, Y.: Infinite random matrices and ergodic decomposition of finite and infinite Hua–Pickrell measures. Adv. Math. 308, 1209–1268 (2017)
Rohlin, V.A.: On the fundamental ideas of measure theory. Am. Math. Soc. Transl. 1952(71), 55 (1952)
Shirai, T., Takahashi, Y.: Fermion process and Fredholm determinant. In: Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999), vol. 7 of International Society for Analysis, its Applications and Computation, pp. 15–23. Kluwer Academic Publishers, Dordrecht (2000)
Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. J. Funct. Anal. 205(2), 414–463 (2003)
Soshnikov, A.: Determinantal random point fields. Uspekhi Mat. Nauk 55(5(335)), 107–160 (2000)
Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)
Acknowledgements
We are deeply grateful to Alexei Klimenko for useful discussions and very helpful comments. The research of A. Bufetov and S. Fan on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under Grant agreement No 647133 (ICHAOS). A. Bufetov has also been funded by the Grant MD 5991.2016.1 of the President of the Russian Federation, by the Russian Academic Excellence Project ‘5-100’ and by the Chaire Gabriel Lamé at the Chebyshev Laboratory of the SPbSU, a joint initiative of the French Embassy in the Russian Federation and the Saint-Petersburg State University. Y. Qiu is supported by the Grant IDEX UNITI-ANR-11-IDEX-0002-02, financed by Programme “Investissements d’Avenir” of the Government of the French Republic managed by the French National Research Agency. Part of this work was carried out at the Institut Henri Poincaré and at the Centre international de rencontres mathématiques in the framework of the CIRM “recherche en petits groupes” programme. We are deeply grateful to these institutions for their warm hospitality.
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Bufetov, A.I., Fan, S. & Qiu, Y. Equivalence of Palm measures for determinantal point processes governed by Bergman kernels. Probab. Theory Relat. Fields 172, 31–69 (2018). https://doi.org/10.1007/s00440-017-0803-z
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DOI: https://doi.org/10.1007/s00440-017-0803-z
Keywords
- Bergman kernel
- Determinantal point process
- Conditional measure
- Deletion and insertion tolerance
- Palm equivalence
- Monotone coupling
Mathematics Subject Classification
- Primary 60G55
- Secondary 32A36