×

Equivalence of palm measures for determinantal point processes governed by Bergman kernels. (English) Zbl 1429.60047

Summary: 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 \(U\).

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
32A36 Bergman spaces of functions in several complex variables
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Balk, M.B.: Polyanalytic functions and their generalizations. In: Complex Analysis, I. Encyclopaedia Mathematical Science, vol. 85, pp. 195-253. Springer, Berlin (1997)
[2] Bergman, S.: The Kernel Function and Conformal Mapping, vol. 5. American Mathematical Society, Providence RI (1970) · Zbl 0208.34302
[3] Bufetov, AI, Infinite determinantal measures and the ergodic decomposition of infinite pickrell measures. I. construction of infinite determinantal measures, Izv. Ross. Akad. Nauk Ser. Mat., 79, 18-64, (2015) · Zbl 1367.37003 · doi:10.4213/im8383
[4] Bufetov, A.I.: Quasi-symmetries of determinantal point processes. arXiv:1409.2068
[5] Bufetov, AI; Qiu, Y., Determinantal point processes associated with Hilbert spaces of holomorphic functions, Commun. Math. Phys., 351, 1-44, (2017) · Zbl 1406.60073 · doi:10.1007/s00220-017-2840-y
[6] Bufetov, A.I., Qiu, Y., Shamov, A.: Kernels of conditional determinantal measures. arXiv:1612.06751
[7] 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 · Zbl 1026.60061
[8] Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Relat. Fields, pp. 1-23 (2014)
[9] Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. To appear in Duke Math. J. arXiv:1211.3506 · Zbl 1405.60067
[10] Haimi, A.; Hedenmalm, H., The polyanalytic Ginibre ensembles, J. Stat. Phys., 153, 10-47, (2013) · Zbl 1278.82068 · doi:10.1007/s10955-013-0813-x
[11] Haimi, A.; Hedenmalm, H., Asymptotic expansion of polyanalytic Bergman kernels, J. Funct. Anal., 267, 4667-4731, (2014) · Zbl 1310.30040 · doi:10.1016/j.jfa.2014.09.002
[12] Holroyd, AE; Soo, T., Insertion and deletion tolerance of point processes, Electron. J. Probab., 18, 24, (2013) · Zbl 1291.60101
[13] Hough, JB; Krishnapur, M.; Peres, Y.; Virág, B., Determinantal processes and independence, Probab. Surv., 3, 206-229, (2006) · Zbl 1189.60101 · doi:10.1214/154957806000000078
[14] Kallenberg, O.: Random Measures, 4th edn. Akademie-Verlag, Berlin (1986) · Zbl 0345.60032
[15] Khintchine, A. Ya, Mathematical methods of queuing theory, Proceedings of the Steklov Institute, 49, 3-122, (1955)
[16] Krantz, S.G.: Geometric Analysis of the Bergman Kernel and Metric, Graduate Texts in Mathematics, vol. 268. Springer, New York (2013) · Zbl 1281.32004 · doi:10.1007/978-1-4614-7924-6
[17] Lindvall, T., On strassen’s theorem on stochastic domination, Electron. Commun. Probab., 4, 51-59, (1999) · Zbl 0938.60013 · doi:10.1214/ECP.v4-1005
[18] Lyons, R.: Determinantal probability: basic properties and conjectures. In: Proceedings of the International Congress of Mathematicians 2014, vol. IV, pp. 137-161. Seoul (2014) · Zbl 1373.60087
[19] Macchi, O., The coincidence approach to stochastic point processes, Adv. Appl. Probab., 7, 83-122, (1975) · Zbl 0366.60081 · doi:10.2307/1425855
[20] Osada, H.; Shirai, T., Absolute continuity and singularity of palm measures of the Ginibre point process, Probab. Theory Relat. Fields, 165, 725-770, (2016) · Zbl 1344.60042 · doi:10.1007/s00440-015-0644-6
[21] Palm, C., Intensitätsschwankungen im fernsprechverkehr, Ericsson Tech., 44, 1-189, (1943) · Zbl 0063.06088
[22] Peres, Y.; Virág, B., Zeros of the iid Gaussian power series: a conformally invariant determinantal process, Acta Math., 194, 1-35, (2005) · Zbl 1099.60037 · doi:10.1007/BF02392515
[23] Qiu, Y., Infinite random matrices and ergodic decomposition of finite and infinite hua-pickrell measures, Adv. Math., 308, 1209-1268, (2017) · Zbl 1407.60011 · doi:10.1016/j.aim.2017.01.003
[24] Rohlin, VA, On the fundamental ideas of measure theory, Am. Math. Soc. Transl., 1952, 55, (1952)
[25] 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) · Zbl 1036.60045
[26] Shirai, T.; Takahashi, Y., Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes, J. Funct. Anal., 205, 414-463, (2003) · Zbl 1051.60052 · doi:10.1016/S0022-1236(03)00171-X
[27] Soshnikov, A., Determinantal random point fields, Uspekhi Mat. Nauk, 55, 107-160, (2000) · Zbl 0991.60038 · doi:10.4213/rm321
[28] Strassen, V., The existence of probability measures with given marginals, Ann. Math. Stat., 36, 423-439, (1965) · Zbl 0135.18701 · doi:10.1214/aoms/1177700153
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.