## Electronic Journal of Probability

### Complex Determinantal Processes and $H1$ Noise

#### Abstract

For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes $\mathcal Z_\rho$ with intensity $\rho d\nu$, where $\nu$ is the corresponding invariant measure. We show that as $\rho\to\infty$, after centering, these processes converge to invariant $H^1$ noise. More precisely, for all functions $f\in H^1(\nu) \cap L^1(\nu)$ the distribution of $\sum_{z\in \mathcal Z} f(z)-\frac{\rho}{\pi} \int f d \nu$ converges to Gaussian with mean zero and variance $\frac{1}{4 \pi} \|f\|_{H^1}^2$.

#### Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 45, 1238-1257.

Dates
Accepted: 9 October 2007
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464818516

Digital Object Identifier
doi:10.1214/EJP.v12-446

Mathematical Reviews number (MathSciNet)
MR2346510

Zentralblatt MATH identifier
1127.60048

Rights

#### Citation

Rider, Brian; Virag, Balint. Complex Determinantal Processes and $H1$ Noise. Electron. J. Probab. 12 (2007), paper no. 45, 1238--1257. doi:10.1214/EJP.v12-446. https://projecteuclid.org/euclid.ejp/1464818516

#### References

• Borodin, A, Okounkov, A., and Olshanski, G. (2000). Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13, 481-515.
• Burton, R., and Pemantle, R. (1993). Local characteristics, entropy, and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21. 1329-1371.
• Costin, O. and Lebowitz, J. (1995). Gaussian fluctuations in random matrices. Phys. Review Letters 75, 69-72.
• Diaconis, P. (2003). Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Linear functionals of eigenvalues of random matrices. Bull. Amer. Math. Soc. 40, 155-178.
• Diaconis, P. and Evans, S.N. (2001). Linear functionals of eigenvalues of random matrices. Trans. AMS 353, 2615-2633.
• Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440-449.
• Helgason, S. Geometric analysis on symmetric spaces, Volume 39 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1994.
• Hough, J., Krishnapur, M., Peres, Y., and Virag, B. (2006) Determinantal processes and independence. Probab. Surveys. 3, 206-229.
• Krishnapur, M. (2006) Zeros of Random Analytic Functions. Ph. D. thesis. Univ. of Ca., Berkeley. arXiv:math.PR/0607504.
• Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Etudes Sci. 98, 167-212.
• Lyons, R., and Steif, J. (2003). Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination. Duke Math J. 120, 515-575.
• Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7, 83-122.
• Peres, Y. and Virag, B. (2005). Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process. Acta. Math., 194, 1-35.
• Rider, B., and Virag, B. (2007). The noise in the circular law and the Gaussian free field. Int. Math. Res. Notices, 2007, article ID rnm006, 32pp.
• Sheffield, S. (2005). Gaussian free fields for mathematicians. Preprint..
• Shirai, T. and Takahashi, Y. (2003) Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their ergodic and Gibbs properties. Ann. Probab. 31 no. 3, 1533-1564.
• Sodin, M. and Tsielson, B. (2006) Random complex zeroes, I. Asymptotic normalitiy. Israel Journal of Mathematics 152, 125-149.
• Soshnikov, A. (2000) Determinantal random fields. Russian Math. Surveys 55, no. 5, 923-975.
• Soshnikov, A (2000). Central Limit Theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28, 1353-1370.
• Soshnikov, A (2002). Gaussian limits for determinantal random point fields. Ann. Probab. 30, 171-181.