Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Zeros of a random analytic function approach perfect spacing under repeated differentiation


Authors: Robin Pemantle and Sneha Subramanian
Journal: Trans. Amer. Math. Soc. 369 (2017), 8743-8764
MSC (2010): Primary 30B20, 60G55; Secondary 30C15
DOI: https://doi.org/10.1090/tran/6929
Published electronically: June 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider an analytic function whose zero set forms a unit intensity Poisson process on the real line. We show that repeated differentiation causes the zero set to converge in distribution to a random translate of the integers.


References [Enhancements On Off] (What's this?)

  • [BBL09] Julius Borcea, Petter Brändén, and Thomas M. Liggett, Negative dependence and the geometry of polynomials, J. Amer. Math. Soc. 22 (2009), no. 2, 521-567. MR 2476782, https://doi.org/10.1090/S0894-0347-08-00618-8
  • [Bre89] Francesco Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989), no. 413, viii+106. MR 963833, https://doi.org/10.1090/memo/0413
  • [CC95] Thomas Craven and George Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 2 (1995), no. 4, 420-441. MR 1376305, https://doi.org/10.4310/MAA.1995.v2.n4.a4
  • [Con83] Brian Conrey, Zeros of derivatives of Riemann's $ \xi $-function on the critical line, J. Number Theory 16 (1983), no. 1, 49-74. MR 693393, https://doi.org/10.1016/0022-314X(83)90031-8
  • [Dur10] Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010. MR 2722836
  • [FR05] David W. Farmer and Robert C. Rhoades, Differentiation evens out zero spacings, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3789-3811. MR 2146650, https://doi.org/10.1090/S0002-9947-05-03721-9
  • [LM74] Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zetafunction, Acta Math. 133 (1974), 49-65. MR 0417074
  • [Maj99] Péter Major, The limit behavior of elementary symmetric polynomials of i.i.d.random variables when their order tends to infinity, Ann. Probab. 27 (1999), no. 4, 1980-2010. MR 1742897, https://doi.org/10.1214/aop/1022677557
  • [Mar49] Morris Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Mathematical Surveys, No. 3, American Mathematical Society, New York, N. Y., 1949. MR 0031114
  • [MS82] T. F. Móri and G. J. Székely, Asymptotic behaviour of symmetric polynomial statistics, Ann. Probab. 10 (1982), no. 1, 124-131. MR 637380
  • [Pem12] Robin Pemantle, Hyperbolicity and stable polynomials in combinatorics and probability, Current developments in mathematics, 2011, Int. Press, Somerville, MA, 2012, pp. 57-123. MR 3098077
  • [Sto26] A. Stoyanoff, Sur un théorem de M. Marcel Riesz, Nouvelles Annales de Mathematique 1 (1926), 97-99.
  • [Sub14] Sneha Dey Subramanian, Zeros, critical points, and coefficients of random functions, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)-University of Pennsylvania. MR 3251112

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30B20, 60G55, 30C15

Retrieve articles in all journals with MSC (2010): 30B20, 60G55, 30C15


Additional Information

Robin Pemantle
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Phildelphia, Pennsylvania 19104
Email: pemantle@math.upenn.edu

Sneha Subramanian
Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
Address at time of publication: Data Scientist, Videa, 3390 Peachtree Road NE, Suite 400, Atlanta, Georgia 30326
Email: sneha.subramanian@videa.tv

DOI: https://doi.org/10.1090/tran/6929
Keywords: Poisson, coefficient, saddle point, lattice, Cauchy integral, random series, translation-invariant
Received by editor(s): October 5, 2014
Received by editor(s) in revised form: March 1, 2016
Published electronically: June 27, 2017
Additional Notes: The first author’s research was supported by NSF grant DMS-1209117
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society