## Abstract

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.

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## Notes

This question was asked to A. N. by Alexei Borodin in a private communication.

The authors are grateful to Brad Rodgers for many insightful discussions on the subject.

This idea was suggested to us by Brad Rodgers.

## References

Aizenman, M., Warzel, S.: On the ubiquity of the Cauchy distribution in spectral problems (2013). arXiv:1312.7769

Bump, D., Gamburd, A.: On the averages of characteristic polynomials from classical groups. Commun. Math. Phys.

**265**(1), 227–274 (2006)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

Bourgade, P., Hughes, C.-P., Nikeghbali, A., Yor, M.: The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J.

**145**(1), 45–69 (2008)Bourgade, P., Najnudel, J., Nikeghbali, A.: A unitary extension of virtual permutations. IMRN

**2013**(18), 4101–4134 (2012)Borodin, A., Olshanski, G., Strahov, E.: Giambelli compatible point processes. Adv. Appl. Math.

**37**(2), 209–248 (2006)Bourgade, P.: Mesoscopic fluctuations of the zeta zeros. Probab. Theory Related Fields

**148**(3–4), 479–500 (2010)Borodin, A., Strahov, E.: Averages of charactersitic polynomials in random matrix theory. Commun. Pure Appl. Math.

**59**(2), 161–253 (2006)Conrey, B., Farmer, D.-W., Zirnbauer, M.-R.: Autocorrelation of ratios of \(L\)-functions. Commun. Number Theory Phys.

**2**(3), 593–636 (2008)Costin, O., Lebowitz, J.: Gaussian fluctuations in random matrices. Phys. Rev. Lett.

**75**(1), 69–72 (1995)Conrey, B., Snaith, N.: Applications of the L-functions ratios conjectures. Proc. Lond. Math. Soc.

**94**(3), 594–646 (2007)Conrey, B., Snaith, N.: Correlation of eigenvalues and riemann zeros. Commun. Number Theory Phys.

**2**(3), 477–536 (2008)Conrey, B., Snaith, N.: In support of n-correlation. Commun. Math. Phys.

**330**(2), 639–653 (2014)Farmer, D.-W., Gonek, S.M., Lee, Y., Lester, S.J.: Mean values of \(\zeta ^{^{\prime }}/\zeta (s)\), correlations of zeros and the distribution of almost primes. Q. J. Math.

**64**(4), 1057–1089 (2013)Fyodorov, Y.-V., Strahov, E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A Math. Gen.

**36**, 3203–3214 (2003)Goldston, D.A., Gonek, S.M., Montgomery, H.L.: Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals. J. Reine Angew. Math.

**537**, 105–126 (2001)Hughes, C.-P., Keating, J.-P., O’Connell, N.: On the characteristic polynomial of a random unitary matrix. Commun. Math. Phys.

**220**, 429–451 (2001)Hughes, C.-P.: On the characteristic polynomial of a random unitary matrix and the riemann zeta function. PhD Thesis (2001)

Keating, J.-P., Snaith, N.: Random matrix theory and \(\zeta (1/2 + it)\). Commun. Math. Phys.

**214**, 57–89 (2000)Meckes, E.-S., Meckes, M.-W.: Spectral measures of powers of random matrices. Electron. Commun. Probab.

**18**(78), 13 (2013)Maples, K., Najnudel, J., Nikeghbali, A.: Limit operators for circular ensembles (2013). arXiv:1304.3757

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)

Rodgers, B.: Tail bounds for counts of zeros and eigenvalues, and an application to ratios (2015). arXiv:1502.05658

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.)Strahov., E., Fyodorov, Y.-V.: On universality of correlation functions of characteristic polynomials: Riemann–Hilbert approach. Commun. Math. Phys.

**241**, 343–382 (2003)Soshnikov, A.: The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab.

**28**(3), 1353–1370 (2000)Soshnikov, A.: Gaussian limit for determinantal random point fields. Ann. Probab.

**30**(1), 171–187 (2002)Titchmarsh, E.-C.: The theory of the Riemann zeta-function, 2nd edn. The Clarendon Press, Oxford University Press, New York (1986). (edited and with a preface by D. R. Heath-Brown)

## Acknowledgments

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.

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Chhaibi, R., Najnudel, J. & Nikeghbali, A. The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios.
*Invent. math.* **207**, 23–113 (2017). https://doi.org/10.1007/s00222-016-0669-1

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DOI: https://doi.org/10.1007/s00222-016-0669-1