Fluctuations of the Process of Moduli for the Ginibre and Hyperbolic Ensembles
DOI:
https://doi.org/10.15407/mag19.01.074Анотація
Ми дослiджуємо точковий процес модулiв ансамблю Жiнiбра та гiперболiчного ансамблю. Ми доводимо, що вдалинi вiд початку координат i вiдносно певної шкали цi процеси виявляють пуассоновi i гаусовi флуктуацiї. Серед можливих гаусових флуктуацiй ми можемо знайти бiлий шум, а також гаусовi флуктуацiї з нетривiальною коварiацiєю на деяких шкалах.
Mathematical Subject Classification 2020: 60G55, 30B20, 30H20
Ключові слова:
ансамбль Жiнiбра, гiперболiчний ансамбль, процес модулей, нормальний розподіл, бiлий шум, пуассонiв точковий процесПосилання
A. Borodin, Determinantal point processes, Oxford Handbook of Random Matrix Theory, Oxford Univ. Press, Oxford, 2011.
A.I. Bufetov, The conditional measures for the determinantal point process with the Bergman kernel, preprint, https://arxiv.org/abs/2112.15557
A.I. Bufetov and A.V. Dymov, A functional limit theorem for the sine-process, Int. Math. Res. Not. IMRN (2019), No. 1, 249--319. https://doi.org/10.1093/imrn/rny104
A.I. Bufetov and Y. Qiu, Determinantal point processes associated with Hilbert spaces of holomorphic functions, Comm. Math. Phys. 351 (2017), 1--44. https://doi.org/10.1007/s00220-017-2840-y
D.J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. II, General theory and structure. Second edition. Probability and its Applications (New York), Springer, New York, 2008. https://doi.org/10.1007/978-0-387-49835-5
S. Ghosh, Determinantal processes and completeness of random exponentials: the critical case, Probab. Theory Related Fields 163 (2015), 643--665. https://doi.org/10.1007/s00440-014-0601-9
S. Ghosh and Y. Peres, Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues, Duke Math. J. 166 (2017), 1789--1858. https://doi.org/10.1215/00127094-2017-0002
J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Mathematical Phys. 6 (1965), 440--449. https://doi.org/10.1063/1.1704292
J.B. Hough, M. Krishnapur, Y. Peres, and B. Virág, Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series, 51. American Mathematical Society, Providence, RI, 2009. https://doi.org/10.1090/ulect/051
Y.G. Kondratiev and T. Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), 201--233. https://doi.org/10.1142/S0219025702000833
E. Kostlan, On the spectra of Gaussian matrices, Linear Algebra Appl. 162/164 (1992), 385--388. https://doi.org/10.1016/0024-3795(92)90386-O
M. Krishnapur, From random matrices to random analytic functions, Ann. Probab. 37 (2009), 314--346. https://doi.org/10.1214/08-AOP404
A. Lenard, Correlation functions and the uniqueness of the state in classical statistical mechanics, Comm. Math. Phys. 30 (1973), 35--44. https://doi.org/10.1007/BF01646686
O. Macchi, The coincidence approach to stochastic point processes, Advances in Appl. Probability. 7 (1975), 83--122. https://doi.org/10.1017/S0001867800040313
Y. Peres and B. Virág, Zeros of the i.i.d. Gaussian power series: A conformally invariant determinantal process, Acta Math. 194 (2005), 1--35. https://doi.org/10.1007/BF02392515
T. Shirai and Y. Takahashi, Fermion process and Fredholm determinant, Proceedings of the Second ISAAC Congress, Vol. 1 (Fukuoka, 1999). Int. Soc. Anal. Appl. Comput., 7, Kluwer Acad. Publ., Dordrecht, 2000, 15--23. https://doi.org/10.1007/978-1-4613-0269-8_3
A. Soshnikov, Determinantal random point fields, Russian Math. Surveys. 55 (2000), 923--975. https://doi.org/10.1070/RM2000v055n05ABEH000321
A. Soshnikov, Gaussian limit for determinantal random point fields, Ann. Probab. 30 (2002), 171--187. https://doi.org/10.1214/aop/1020107764
