## The Annals of Probability

### Asymptotic distribution of complex zeros of random analytic functions

#### Abstract

Let $\xi_{0},\xi_{1},\ldots$ be independent identically distributed complex-valued random variables such that $\mathbb{E}\log(1+|\xi_{0}|)<\infty$. We consider random analytic functions of the form $\mathbf{G}_{n}(z)=\sum_{k=0}^{\infty}\xi_{k}f_{k,n}z^{k},$ where $f_{k,n}$ are deterministic complex coefficients. Let $\mu_{n}$ be the random measure counting the complex zeros of $\mathbf{G}_{n}$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}\mu_{n}$ converges in probability to some deterministic measure $\mu$ which is characterized in terms of the Legendre–Fenchel transform of $u$. The limiting measure $\mu$ does not depend on the distribution of the $\xi_{k}$’s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.

#### Article information

Source
Ann. Probab. Volume 42, Number 4 (2014), 1374-1395.

Dates
First available in Project Euclid: 3 July 2014

https://projecteuclid.org/euclid.aop/1404394067

Digital Object Identifier
doi:10.1214/13-AOP847

Mathematical Reviews number (MathSciNet)
MR3262481

Zentralblatt MATH identifier
1295.30008

#### Citation

Kabluchko, Zakhar; Zaporozhets, Dmitry. Asymptotic distribution of complex zeros of random analytic functions. Ann. Probab. 42 (2014), no. 4, 1374--1395. doi:10.1214/13-AOP847. https://projecteuclid.org/euclid.aop/1404394067

#### References

• [1] Arnold, L. (1966). Über die Nullstellenverteilung zufälliger Polynome. Math. Z. 92 12–18.
• [2] Bharucha-Reid, A. T. and Sambandham, M. (1986). Random Polynomials. Academic Press, Orlando, FL.
• [3] Bleher, P. and Di, X. (2004). Correlations between zeros of non-Gaussian random polynomials. Int. Math. Res. Not. IMRN 2004 2443–2484.
• [4] Bloom, T. and Shiffman, B. (2007). Zeros of random polynomials on $\mathbf{C}^{m}$. Math. Res. Lett. 14 469–479.
• [5] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1–89.
• [6] Edelman, A. and Kostlan, E. (1995). How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) 32 1–37.
• [7] Esseen, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrsch. verw. Gebiete 9 290–308.
• [8] Farahmand, K. (1998). Topics in Random Polynomials. Pitman Research Notes in Mathematics Series 393. Longman, Harlow.
• [9] Forrester, P. J. and Honner, G. (1999). Exact statistical properties of the zeros of complex random polynomials. J. Phys. A 32 2961–2981.
• [10] Hammersley, J. M. (1956). The zeros of a random polynomial. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 19541955, vol. II, 89–111. Univ. California Press, Berkeley and Los Angeles.
• [11] Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators I. Grundlehren der Mathematischen Wissenschaften 256. Springer, Berlin.
• [12] Hörmander, L. (1983). The Analysis of Linear Partial Differential Operators II. Grundlehren der Mathematischen Wissenschaften 257. Springer, Berlin.
• [13] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. Amer. Math. Soc., Providence, RI.
• [14] Hughes, C. P. and Nikeghbali, A. (2008). The zeros of random polynomials cluster uniformly near the unit circle. Compos. Math. 144 734–746.
• [15] Ibragimov, I. and Zeitouni, O. (1997). On roots of random polynomials. Trans. Amer. Math. Soc. 349 2427–2441.
• [16] Ibragimov, I. A. and Zaporozhets, D. N. (2013). On distribution of zeros of random polynomials in complex plane. In Prokhorov and Contemporary Probability Theory (A. N. Shiryaev, S. R. S. Varadhan and E. L. Presman, eds.). Springer Proceedings in Mathematics and Statistics 33 303–324. Springer, Berlin.
• [17] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
• [18] Krishnapur, M. and Virág, B. (2014). The Ginibre ensemble and Gaussian analytic functions. Int. Math. Res. Not. IMRN. 2014 1441–1464.
• [19] Ledoan, A., Merkli, M. and Starr, S. (2012). A universality property of Gaussian analytic functions. J. Theoret. Probab. 25 496–504.
• [20] Levin, B. J. (1964). Distribution of Zeros of Entire Functions. Translations of Mathematical Monographs 5. Amer. Math. Soc., Providence, RI.
• [21] Littlewood, J. E. and Offord, A. C. (1938). On the number of real roots of a random algebraic equation. J. Lond. Math. Soc. (2) 13 288–295.
• [22] Littlewood, J. E. and Offord, A. C. (1939). On the number of real roots of a random algebraic equation. II. Math. Proc. Cambridge Philos. Soc. 35 133–148.
• [23] Littlewood, J. E. and Offord, A. C. (1945). On the distribution of the zeros and $a$-values of a random integral function. I. J. Lond. Math. Soc. (2) 20 130–136.
• [24] Littlewood, J. E. and Offord, A. C. (1948). On the distribution of zeros and $a$-values of a random integral function. II. Ann. of Math. (2) 49 885–952. Errata: 50 990–991 (1949).
• [25] Offord, A. C. (1965). The distribution of the values of an entire function whose coefficients are independent random variables. Proc. Lond. Math. Soc. (3) 14a 199–238.
• [26] Offord, A. C. (1995). The distribution of the values of an entire function whose coefficients are independent random variables. II. Math. Proc. Cambridge Philos. Soc. 118 527–542.
• [27] Schehr, G. and Majumdar, S. N. (2009). Condensation of the roots of real random polynomials on the real axis. J. Stat. Phys. 135 587–598.
• [28] Shepp, L. A. and Vanderbei, R. J. (1995). The complex zeros of random polynomials. Trans. Amer. Math. Soc. 347 4365–4384.
• [29] Shiffman, B. and Zelditch, S. (1999). Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200 661–683.
• [30] Shiffman, B. and Zelditch, S. (2003). Equilibrium distribution of zeros of random polynomials. Int. Math. Res. Not. IMRN 2003 25–49.
• [31] Shmerling, E. and Hochberg, K. J. (2002). Asymptotic behavior of roots of random polynomial equations. Proc. Amer. Math. Soc. 130 2761–2770 (electronic).
• [32] Sodin, M. (2005). Zeroes of Gaussian analytic functions. In European Congress of Mathematics 445–458. Eur. Math. Soc., Zürich.
• [33] Sodin, M. and Tsirelson, B. (2004). Random complex zeroes. I. Asymptotic normality. Israel J. Math. 144 125–149.
• [34] Szegő, G. (1924). Über eine Eigenschaft der Exponentialreihe. Sitzungsber. Berl. Math. Ges. 23 50–64.
• [35] Tao, T. and Vu, V. (2010). Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 2023–2065. With an appendix by Manjunath Krishnapur.
• [36] Šparo, D. I. and Šur, M. G. (1962). On the distribution of roots of random polynomials. Vestnik Moskov. Univ. Ser. I Mat. Meh. 1962 40–43.