## Electronic Journal of Probability

### Hole probabilities for $\beta$-ensembles and determinantal point processes in the complex plane

#### Abstract

We compute the exact decay rate of the hole probabilities for $\beta$-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory for both processes.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 48, 21 pp.

Dates
Accepted: 7 May 2018
First available in Project Euclid: 1 June 2018

https://projecteuclid.org/euclid.ejp/1527818426

Digital Object Identifier
doi:10.1214/18-EJP176

Mathematical Reviews number (MathSciNet)
MR3814242

Zentralblatt MATH identifier
06924660

Subjects
Primary: 60G55: Point processes

#### Citation

Adhikari, Kartick. Hole probabilities for $\beta$-ensembles and determinantal point processes in the complex plane. Electron. J. Probab. 23 (2018), paper no. 48, 21 pp. doi:10.1214/18-EJP176. https://projecteuclid.org/euclid.ejp/1527818426

#### References

• [AOC12] Yacin Ameur and Joaquim Ortega-Cerdà. Beurling–landau densities of weighted fekete sets and correlation kernel estimates. Journal of Functional Analysis, 263(7):1825–1861, 2012.
• [AR16] Kartick Adhikari and Nanda Kishore Reddy. Hole probabilities for finite and infinite ginibre ensembles. Int Math Res Notices, no. 21, 6694–6730, 2017.
• [AS13] Gernot Akemann and Eugene Strahov. Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. J. Stat. Phys., 151(6):987–1003, 2013.
• [ASZ14] Scott N Armstrong, Sylvia Serfaty, and Ofer Zeitouni. Remarks on a constrained optimization problem for the ginibre ensemble. Potential Analysis, 41(3):945–958, 2014.
• [ATW14] Romain Allez, Jonathan Touboul, and Gilles Wainrib. Index distribution of the Ginibre ensemble. J. Phys. A, 47(4):042001, 8, 2014.
• [Blo14] Thomas Bloom. Large deviation for outlying coordinates in $\beta$ ensembles. J. Approx. Theory, 180:1–20, 2014.
• [BLW08] Len Bos, Norm Levenberg, and Shayne Waldron. On the spacing of Fekete points for a sphere, ball or simplex. Indag. Math. (N.S.), 19(2):163–176, 2008.
• [BNPS16] Jeremiah Buckley, Alon Nishry, Ron Peled, and Mikhail Sodin. Hole probability for zeroes of gaussian taylor series with finite radii of convergence. arXiv preprint arXiv:1602.03076, 2016.
• [DVJ08] D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. II. Probability and its Applications (New York). Springer, New York, second edition, 2008. General theory and structure.
• [Gin65] Jean Ginibre. Statistical ensembles of complex, quaternion, and real matrices. J. Mathematical Phys., 6:440–449, 1965.
• [GN16] Subhroshekhar Ghosh and Alon Nishry. Gaussian complex zeros on the hole event: the emergence of a forbidden region. arXiv preprint arXiv:1609.00084, 2016.
• [HKPV09] J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág. Zeros of Gaussian analytic functions and determinantal point processes, volume 51 of University Lecture Series. American Mathematical Society, Providence, RI, 2009.
• [HM13] Håkan Hedenmalm and Nikolai Makarov. Coulomb gas ensembles and Laplacian growth. Proc. Lond. Math. Soc. (3), 106(4):859–907, 2013.
• [HMS11] H. J. Haubold, A. M. Mathai, and R. K. Saxena. Mittag-Leffler functions and their applications. J. Appl. Math., pages Art. ID 298628, 51, 2011.
• [Kos92] Eric Kostlan. On the spectra of Gaussian matrices. Linear Algebra Appl., 162/164:385–388, 1992. Directions in matrix theory (Auburn, AL, 1990).
• [Nis10] Alon Nishry. Asymptotics of the hole probability for zeros of random entire functions. Int. Math. Res. Not. IMRN, (15):2925–2946, 2010.
• [Nis11] Alon Nishry. The hole probability for Gaussian entire functions. Israel J. Math., 186:197–220, 2011.
• [Nis12] Alon Nishry. Hole probability for entire functions represented by Gaussian Taylor series. J. Anal. Math., 118(2):493–507, 2012.
• [Shi06] Tomoyuki Shirai. Large deviations for the fermion point process associated with the exponential kernel. J. Stat. Phys., 123(3):615–629, 2006.
• [ST97] Edward B. Saff and Vilmos Totik. Logarithmic potentials with external fields, volume 316 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom.
• [ST05] Mikhail Sodin and Boris Tsirelson. Random complex zeroes. III. Decay of the hole probability. Israel J. Math., 147:371–379, 2005.