On the maximum of the C\(\beta\)E field. (English) Zbl 1457.60008

In this paper, the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (\(C\beta E\)) are investigated. More precisely, assuming that \(X_n\) is this characteristic polynomial and \(\mathbb{U}\) is the unit circle, it is proved that \[\sup\limits_{z\in\mathbb{U}}\mathrm{Re} \log{X_n(z)}=\sqrt{\frac{2}{\beta}}\left(\log n-{\frac{3}{4}} \log{\log{n}}+\mathcal{O}(1)\right)\] as well as an analogous statement for the imaginary part. The notation \(\mathcal{O}(1)\) means that the corresponding family of random variables, indexed by \(n\), is tight. This answers a conjecture of [Y. Fyodorov et al., “Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function”, Phys. Rev. Lett., 108, No. 17, Article ID 170601, 5 p. (2012; doi:10.1103/PhysRevLett.108.170601); Y. Fyodorov and J. Keating, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 372, No. 2007, Article ID 20120503, 32 p. (2014; Zbl 1330.82028)]), originally formulated for the \(\beta=2\) case, which corresponds to the circular unitary ensemble (\(CUE\)) field.


60B20 Random matrices (probabilistic aspects)
60G70 Extreme value theory; extremal stochastic processes
15B52 Random matrices (algebraic aspects)


Zbl 1330.82028
Full Text: DOI arXiv Euclid


[1] E. Aïdékon, Convergence in law of the minimum of a branching random walk, Ann. Probab. 41 (2013), 1362–1426. · Zbl 1285.60086
[2] E. Aïdékon, J. Berestycki, E. Brunet, and Z. Shi, Branching Brownian motion seen from its tip, Probab. Theory Related Fields 157 (2013), 405–451. · Zbl 1284.60154
[3] E. Aïdékon and Z. Shi, Weak convergence for the minimal position in a branching random walk: A simple proof, Period. Math. Hungar. 61 (2010), 43–54.
[4] E. Aïdékon and Z. Shi, The Seneta-Heyde scaling for the branching random walk, Ann. Probab. 42 (2014), 959–993. · Zbl 1304.60092
[5] L.-P. Arguin, D. Belius, and P. Bourgade, Maximum of the characteristic polynomial of random unitary matrices, Comm. Math. Phys. 349 (2017), 703–751. · Zbl 1371.15036
[6] L.-P. Arguin, D. Belius, P. Bourgade, M. Raziwiłł, and K. Soundararajan, Maximum of the Riemann zeta function on a short interval of the critical line, to appear in Comm. Pure Appl. Math., preprint, arXiv:1612.08575v3 [math.PR].
[7] L.-P. Arguin, D. Belius, and A. J. Harper, Maxima of a randomized Riemann zeta function, and branching random walks, Ann. Appl. Probab. 27 (2017), 178–215. · Zbl 1362.60050
[8] L.-P. Arguin, A. Bovier, and N. Kistler, The extremal process of branching Brownian motion, Probab. Theory Related Fields 157 (2013), 535–574. · Zbl 1286.60045
[9] L.-P. Arguin and O. Zindy, Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field, Ann. Appl. Probab. 24 (2014), 1446–1481. · Zbl 1301.60042
[10] D. Belius and N. Kistler, The subleading order of two dimensional cover times, Probab. Theory Related Fields 167 (2017), 461–552. · Zbl 1365.60071
[11] M. Biskup and O. Louidor, Extreme local extrema of two-dimensional discrete Gaussian free field, Comm. Math. Phys. 345 (2016), 271–304. · Zbl 1347.82007
[12] P. Bourgade, Mesoscopic fluctuations of the zeta zeros, Probab. Theory Related Fields 148 (2010), 479–500. · Zbl 1250.11080
[13] P. Bourgade, C.-P. Hughes, A. Nikeghbali, and M. Yor, The characteristic polynomial of a random unitary matrix: A probabilistic approach, Duke Math. J. 145 (2008), 45–69. · Zbl 1155.15025
[14] P. Bourgade, A. Nikeghbali, and A. Rouault, Circular Jacobi ensembles and deformed Verblunsky coefficients, Int. Math. Res. Not. IMRN 2009, no. 23, 4357–4394. · Zbl 1183.33013
[15] M. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978), 531–581. · Zbl 0361.60052
[16] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983), no. 285. · Zbl 0517.60083
[17] M. Bramson, J. Ding, and O. Zeitouni, Convergence in law of the maximum of nonlattice branching random walk, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), 1897–1924. · Zbl 1355.60066
[18] M. Bramson, J. Ding, and O. Zeitouni, Convergence in law of the maximum of the two-dimensional discrete Gaussian free field, Comm. Pure Appl. Math. 69 (2016), 62–123. · Zbl 1355.60046
[19] M. Bramson and O. Zeitouni, Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field, Comm. Pure Appl. Math. 65 (2012), 1–20. · Zbl 1237.60041
[20] M. J. Cantero, L. Moral, and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29–56. · Zbl 1022.42013
[21] R. Chhaibi, J. Najnudel, and A. Nikeghbali, The circular unitary ensemble and the Riemann zeta function: The microscopic landscape and a new approach to ratios, Invent. Math. 207 (2016), 23–113. · Zbl 1368.11098
[22] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49–62. · Zbl 0807.15015
[23] J. Ding, R. Roy, and O. Zeitouni, Convergence of the centered maximum of log-correlated Gaussian fields, Ann. Probab. 45 (2017), 3886–3928. · Zbl 1412.60058
[24] B. Duplantier and S. Sheffield, Liouville quantum gravity and KPZ, Invent. Math. 185 (2011), 333–393. · Zbl 1226.81241
[25] Y.-V. Fyodorov, G.-A. Hiary, and J.-P. Keating, Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function, Phys. Rev. Lett. 108 (2012), no. 17, art. ID 170601. · Zbl 1267.82055
[26] Y.-V. Fyodorov and J.-P. Keating, Freezing transitions and extreme values: random matrix theory and disordered landscapes, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 20120503. · Zbl 1330.82028
[27] Y. Hu and Z. Shi, Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees, Ann. Probab. 37 (2009), 742–789. · Zbl 1169.60021
[28] C.-P. Hughes, J.-P. Keating, and N. O’Connell, On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 220 (2001), 429–451. · Zbl 0987.60039
[29] T. Jiang and S. Matsumoto, Moments of traces of circular beta-ensembles, Ann. Probab. 43 (2015), 3279–3336. · Zbl 1388.60029
[30] K. Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (1997), 519–545. · Zbl 0883.60010
[31] J.-P. Kahane, Sur le chaos multiplicatif, Ann. Math. Qué. 9 (1985), 105–150. · Zbl 0596.60041
[32] J.-P. Keating and N. C. Snaith, Random matrix theory and \(ζ(1/2+it)\), Comm. Math. Phys. 214 (2000), 57–89. · Zbl 1051.11048
[33] R. Killip and I. Nenciu, Matrix models for circular ensembles, Int. Math. Res. Not. IMRN 2004, no. 50, 2665–2701. · Zbl 1255.82004
[34] R. Killip and M. Stoiciu, Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles, Duke Math. J. 146 (2009), 361–399. · Zbl 1155.81020
[35] N. Kistler, “Derrida’s random energy models: From spin glasses to the extremes of correlated random fields” in Correlated Random Systems: Five Different Methods, Lecture Notes in Math. 2143, Springer, Cham, 2015, 71–120. · Zbl 1338.60231
[36] V. G. Knizhnik, A. M. Polyakov, and A. B. Zamolodchikov, Fractal structure of \(2\)D-quantum gravity, Modern Phys. Lett. A 3 (1988), 819–826.
[37] M. V. Kozlov, The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment, Teor. Verojatnost. i Primenen. 21, no. 4 (1976), 813–825.
[38] E. Lukacs, A characterization of the gamma distribution, Ann. Math. Statist. 26 (1955), 319–324. · Zbl 0065.11103
[39] T. Madaule, Maximum of a log-correlated Gaussian field, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), 1369–1431. · Zbl 1329.60138
[40] T. Madaule, Convergence in law for the branching random walk seen from its tip, J. Theoret. Probab. 30 (2017), 27–63. · Zbl 1368.60089
[41] J. Najnudel, On the extreme values of the Riemann zeta function on random intervals of the critical line, Probab. Theory Relat. Fields, published electronically 4 November 2017.
[42] E. Paquette and O. Zeitouni, The maximum of the CUE field, Int. Math. Res. Not. IMRN, published electronically 8 March 2017. · Zbl 1390.60036
[43] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Grundlehren Math. Wiss. 293, Springer, Berlin, 1999. · Zbl 0917.60006
[44] R. Rhodes and V. Vargas, Gaussian multiplicative chaos and applications: A review, Probab. Surv. 11 (2014), 315–392. · Zbl 1316.60073
[45] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Amer. Math. Soc. Colloq. Publ. 54, Amer. Math. Soc., Providence, 2005. · Zbl 1082.42020
[46] C. Webb, Linear statistics of the circular \(β\)-ensemble, Stein’s method and circular Dyson Brownian motion, Electron. J. Probab. 21 (2016), no. 25. · Zbl 1375.15056
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