The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios
We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called “microscopic” level, that is we consider the characteristic polynomial at points whose distance to 1 has order 1 / n. We prove that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. In order to deal with this issue, we couple all the dimensions n on a single probability space, in such a way that almost sure convergence occurs when n goes to infinity. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form \(\exp (2 i \pi \alpha /n)\) and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the work by Keating and Snaith, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the level of stochastic processes.
We would like to thank Brad Rodgers for very stimulating discussions and A.N. would also like to thank Alexei Borodin for mentioning the problems on ratios of characteristic polynomials at the microscopic scale.
- 1.Aizenman, M., Warzel, S.: On the ubiquity of the Cauchy distribution in spectral problems (2013). arXiv:1312.7769
- 3.Barhoumi, Y., Hughes, C.-P., Najnudel, J., Nikeghbali, A.: On the number of zeros of linear combinations of indepepndent characteristic polynomials of random unitary matrices. arXiv:1301.5144
- 18.Hughes, C.-P.: On the characteristic polynomial of a random unitary matrix and the riemann zeta function. PhD Thesis (2001)Google Scholar
- 21.Maples, K., Najnudel, J., Nikeghbali, A.: Limit operators for circular ensembles (2013). arXiv:1304.3757
- 22.Montgomery, H.L.: The pair correlation of zeros of the zeta function. In: Analytic number theory (Proc. Sympos. Pure Math., vol. XXIV, St. Louis Univ., St. Louis, MO, 1972), pp. 181–193. Amer. Math. Soc., Providence (1973)Google Scholar
- 23.Rodgers, B.: Tail bounds for counts of zeros and eigenvalues, and an application to ratios (2015). arXiv:1502.05658
- 24.Rudnick, Z., Sarnak, P.: Zeros of principal L-functions and random matrix theory. Duke Math. J. 81(2), 269–322 (1996). (a celebration of John F. Nash, Jr.)Google Scholar
- 25.Strahov., E., Fyodorov, Y.-V.: On universality of correlation functions of characteristic polynomials: Riemann–Hilbert approach. Commun. Math. Phys. 241, 343–382 (2003)Google Scholar