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# Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

## Abstract

Let $$\xi _0,\xi _1,\ldots$$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions:

\begin{aligned} P_n(z) := {\left\{ \begin{array}{ll} \sum \nolimits _{k=0}^n \sqrt{\left( {\begin{array}{c}n\\ k\end{array}}\right) } \xi _k z^k &{}\text { (spherical polynomials)},\\ \sum \nolimits _{k=0}^\infty \sqrt{\frac{n^k}{k!}} \xi _k z^k &{}\text { (flat random analytic function)},\\ \sum \nolimits _{k=0}^\infty \sqrt{\left( {\begin{array}{c}n+k-1\\ k\end{array}}\right) } \xi _k z^k &{}\text { (hyperbolic random analytic functions)},\\ \sum \nolimits _{k=0}^n \sqrt{\frac{n^k}{k!}} \xi _k z^k &{}\text { (Weyl polynomials)}. \end{array}\right. } \end{aligned}

We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $$\lim _{n\rightarrow \infty } n^{-1/2}\mathbb {E} N_n[a,b]$$, where $$N_n[a,b]$$ is the number of zeroes of $$P_n$$ in the interval [ab].

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## Notes

1. 1.

A random variable Z has negative binomial distribution $$\text {NBin}(n,p)$$ if $$\mathbb {P}\left[ Z=k\right] = \left( {\begin{array}{c}n+k-1\\ k\end{array}}\right) p^n(1-p)^k$$ for $$k=0,1,\ldots$$.

## References

1. 1.

Angst, J., Poly, G., Viet, H.P.: Universality of the nodal length of bivariate random trigonometric polynomials. arXiv:1610.05360 (2016)

2. 2.

Azaïs, J.-M., Wschebor, M.: Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken (2009). https://doi.org/10.1002/9780470434642

3. 3.

Bleher, P., Di, X.: Correlations between zeros of non-Gaussian random polynomials. Int. Math. Res. Not. 46, 2443–2484 (2004)

4. 4.

Bleher, P., Ridzal, D.: $${\text{ SU }}(1,1)$$ random polynomials. J. Stat. Phys. 106(1–2), 147–171 (2002)

5. 5.

Bloch, A., Pólya, G.: On the roots of certain algebraic equations. Proc. Lond. Math. Soc. 2(33), 102–114 (1931). https://doi.org/10.1112/plms/s2-33.1.102

6. 6.

Bogomolny, E., Bohigas, O., Lebœuf, P.: Distribution of roots of random polynomials. Phys. Rev. Lett. 68(18), 2726–2729 (1992). https://doi.org/10.1103/PhysRevLett.68.2726

7. 7.

Bogomolny, E., Bohigas, O., Leboeuf, P.: Quantum chaotic dynamics and random polynomials. J. Stat. Phys. 85(5–6), 639–679 (1996). https://doi.org/10.1007/BF02199359

8. 8.

Conway, J.B.: Functions of One Complex Variable. Graduate Texts in Mathematics. 11. Springer, New York (1973)

9. 9.

Cramer, H., Leadbetter, M.R.: Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications. Wiley, New York (1967)

10. 10.

Do, Y., Nguyen, H., Vu, V.: Real roots of random polynomials: expectation and repulsion. Proc. Lond. Math. Soc. (3) 111(6), 1231–1260 (2015). https://doi.org/10.1112/plms/pdv055

11. 11.

Do, Y., Nguyen, O., Vu, V.: Roots of random polynomials with coefficients having polynomial growth. arXiv:1507.04994 (2015)

12. 12.

Edelman, A., Kostlan, E.: How many zeros of a random polynomial are real? Bull. Am. Math. Soc. 32(1), 1–37 (1995)

13. 13.

Esseen, C.G.: On the concentration function of a sum of independent random variables. Z. Wahr. Verw. Gebiete 9, 290–308 (1968). https://doi.org/10.1007/BF00531753

14. 14.

Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1966)

15. 15.

Flasche, H.: Expected number of real roots of random trigonometric polynomials. Stoch. Proc. Appl. 127(12), 3928–3942 (2017). https://doi.org/10.1016/j.spa.2017.03.018

16. 16.

Flasche, H., Kabluchko, Z.: Expected number of real zeros of random Taylor series. arXiv:1709.02937 (2017)

17. 17.

Forrester, P.J., Honner, G.: Exact statistical properties of the zeros of complex random polynomials. J. Phys. A 32(16), 2961–2981 (1999)

18. 18.

Hannay, J.H.: Chaotic analytic zero points: exact statistics for those of a random spin state. J. Phys. A 29(5), L101–L105 (1996). https://doi.org/10.1088/0305-4470/29/5/004

19. 19.

Hannay, J.H.: The chaotic analytic function. J. Phys. A 31(49), L755–L761 (1998). https://doi.org/10.1088/0305-4470/31/49/001

20. 20.

Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, volume 51 of University Lecture Series. American Mathematical Society, Providence (2009)

21. 21.

Ibragimov, I., Zaporozhets, D.: On distribution of zeros of random polynomials in complex plane. In: Shiryaev, A., Varadhan, S., Presman, E. (eds.) Prokhorov and Contemporary Probability Theory, pp. 303–323. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-33549-5_18

22. 22.

Ibragimov, I.A., Maslova, N.B.: The mean number of real zeros of random polynomials. I. Coefficients with zero mean. Teor. Verojatnost. i Primenen 16, 229–248 (1971)

23. 23.

Ibragimov, I.A., Maslova, N.B.: The mean number of real zeros of random polynomials. II. Coefficients with a nonzero mean. Teor. Verojatnost. i Primenen. 16, 495–503 (1971)

24. 24.

Iksanov, A., Kabluchko, Z., Marynych, A.: Local universality for real roots of random trigonometric polynomials. Electron. J. Probab. 21, 19 (2016). https://doi.org/10.1214/16-EJP9

25. 25.

Kabluchko, Z., Zaporozhets, D.: Asymptotic distribution of complex zeros of random analytic functions. Ann. Probab. 42(4), 1374–1395 (2014). https://doi.org/10.1214/13-AOP847

26. 26.

Kac, M.: On the average number of real roots of a random algebraic equation. Proc. Lond. Math. Soc. 2(50), 390–408 (1948). https://doi.org/10.1112/plms/s2-50.5.390

27. 27.

Kostlan, E.: On the distribution of roots of random polynomials. In: From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), pp. 419–431. Springer, Berlin (1993)

28. 28.

Ledoan, A., Merkli, M., Starr, S.: A universality property of Gaussian analytic functions. J. Theor. Probab. 25(2), 496–504 (2012)

29. 29.

Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation. J. Lond. Math. Soc. 13, 288–295 (1938). https://doi.org/10.1112/jlms/s1-13.4.288

30. 30.

Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation. II. Proc. Camb. Philos. Soc. 35, 133–148 (1939)

31. 31.

Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation. III. Mat. Sb. Nov. Ser. 12, 277–286 (1943)

32. 32.

Lubinsky, D.S., Pritsker, I.E., Xie, X.: Expected number of real zeros for random linear combinations of orthogonal polynomials. Proc. Am. Math. Soc. 144(4), 1631–1642 (2016)

33. 33.

Lubinsky, D.S., Pritsker, I.E., Xie, X.: Expected number of real zeros for random orthogonal polynomials. Math. Proc. Camb. Philos. Soc. 164(1), 47–66 (2018)

34. 34.

Maslova, N.B.: The distribution of the number of real roots of random polynomials. Teor. Verojatnost. i Primenen. 19, 488–500 (1974a)

35. 35.

Maslova, N.B.: The variance of the number of real roots of random polynomials. Teor. Verojatnost. i Primenen. 19, 36–51 (1974b)

36. 36.

Nguyen, H., Nguyen, O., Vu, V.: On the number of real roots of random polynomials. Commun. Contemp. Math. 18(4), 17 (2016). https://doi.org/10.1142/S0219199715500522

37. 37.

Petrov, V.V.: Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 82. Springer, Berlin (1975)

38. 38.

Pritsker, I.E., Varga, R.S.: The Szegő curve, zero distribution and weighted approximation. Trans. Am. Math. Soc. 349(10), 4085–4105 (1997)

39. 39.

Schehr, G., Majumdar, S.N.: Real roots of random polynomials and zero crossing properties of diffusion equation. J. Stat. Phys. 132(2), 235–273 (2008). https://doi.org/10.1007/s10955-008-9574-3

40. 40.

Shirai, T.: Limit theorems for random analytic functions and their zeros. In: Functions in Number Theory and Their Probabilistic Aspects, RIMS Kôkyûroku Bessatsu, B34, pp. 335–359. Res. Inst. Math. Sci. (RIMS), Kyoto (2012)

41. 41.

Shub, M., Smale, S.: The complexity of Bezout theorem, I–V. In: Cucker, F., Wong, R. (eds.) The Collected Papers of Stephen Smale, vol. 3. Singapore University Press, Singapore (2000)

42. 42.

Sodin, M., Tsirelson, B.: Random complex zeroes. I. Asymptotic normality. Isr. J. Math. 144, 125–149 (2004)

43. 43.

Tao, T., Vu, V.: Local universality of zeroes of random polynomials. Int. Math. Res. Not. 2015(13), 5053–5139 (2015). https://doi.org/10.1093/imrn/rnu084

## Acknowledgements

The support by the SFB 878 “Groups, Geometry and Actions” is gratefully acknowledged.

## Author information

Correspondence to Zakhar Kabluchko.

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Reprints and Permissions

Flasche, H., Kabluchko, Z. Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature. J Theor Probab 33, 103–133 (2020). https://doi.org/10.1007/s10959-018-0843-z

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### Keywords

• Random polynomials
• Random analytic functions
• Spherical polynomials
• Flat analytic function
• Hyperbolic analytic function
• Weyl polynomials
• Real zeroes
• Weak convergence
• Gaussian processes
• Functional limit theorem

### Mathematics Subject Classification (2010)

• Primary: 30C15
• 26C10
• Secondary: 60F99
• 60F17
• 60F05
• 60G15