The Annals of Probability

Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble

Y. V. Fyodorov, B. A. Khoruzhenko, and N. J. Simm

Full-text: Open access


The goal of this paper is to establish a relation between characteristic polynomials of $N\times N$ GUE random matrices $\mathcal{H}$ as $N\to\infty$, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of $D_{N}(z)=-\log|\det(\mathcal{H}-zI)|$ on mesoscopic scales as $N\to\infty$. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, $D_{N}(x)$ gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series.

Article information

Ann. Probab., Volume 44, Number 4 (2016), 2980-3031.

Received: December 2013
Revised: May 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60F17: Functional limit theorems; invariance principles 60F05: Central limit and other weak theorems 15B52: Random matrices

Random matrix theory mesoscopic regime logarithmically correlated fractional Brownian motion generalized processes


Fyodorov, Y. V.; Khoruzhenko, B. A.; Simm, N. J. Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble. Ann. Probab. 44 (2016), no. 4, 2980--3031. doi:10.1214/15-AOP1039.

Export citation


  • [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • [2] Astala, K., Jones, P., Kupiainen, A. and Saksman, E. (2011). Random conformal weldings. Acta Math. 207 203–254.
  • [3] Bacry, E. and Muzy, J. F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449–475.
  • [4] Barral, J. and Mandelbrot, B. B. (2004). Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures. II). In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2. Proc. Sympos. Pure Math. 72 17–52. Amer. Math. Soc., Providence, RI.
  • [5] Bleher, P. M. (2011). Lectures on random matrix models: The Riemann–Hilbert approach. In Random Matrices, Random Processes and Integrable Systems (J. Harnad, ed.). CRM Ser. Math. Phys. 251–349. Springer, New York.
  • [6] Bleher, P. M. and Fokin, V. V. (2006). Exact solution of the six-vertex model with domain wall boundary conditions. Disordered phase. Comm. Math. Phys. 268 223–284.
  • [7] Borodin, A. and Gorin, V. (2013). General beta Jacobi corners process and the Gaussian Free Field. Available at arXiv:1305.3627.
  • [8] Bourgade, P., Erdös, L., Yau, H.-T. and Yin, J. (2014). Fixed energy universality for generalized Wigner matrices. Available at arXiv:1407.5606.
  • [9] Boutet de Monvel, A. and Khorunzhy, A. (1999). Asymptotic distribution of smoothed eigenvalue density. I. Gaussian random matrices. Random Oper. Stoch. Equ. 7 1–22.
  • [10] Boutet de Monvel, A. and Khorunzhy, A. (1999). Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices. Random Oper. Stoch. Equ. 7 149–168.
  • [11] Breuer, J. and Duits, M. (2016). Universality of mesoscopic fluctuations for orthogonal polynomial ensembles. Comm. Math. Phys. 342 491–531.
  • [12] Carpentier, D. and Le Doussal, P. (2001). Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville ane sinh-Gordon models. Phys. Rev. E 63 026110.
  • [13] Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X. (1999). Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 1491–1552.
  • [14] Deift, P. A. (1999). Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. New York Univ., Courant Institute of Mathematical Sciences, New York.
  • [15] Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A 49–62.
  • [16] Doukhan, P, Oppenheim, G. and Taqqu, M. S., eds. (2003). Theory and Applications of Long-Range Dependence. Birkhäuser, Boston, MA.
  • [17] Duits, M. and Johansson, K. (2013). On mesoscopic equilibrium for linear statistics in Dyson’s Brownian motion. Available at arXiv:1312.4295.
  • [18] Ercolani, N. M. and McLaughlin, K. D. T.-R. (2003). Asymptotics of the partition function for random matrices via Riemann–Hilbert techniques and applications to graphical enumeration. Int. Math. Res. Not. 14 755–820.
  • [19] Erdős, L. and Knowles, A. (2015). The Altshuler–Shklovskii formulas for random band matrices I: The unimodular case. Comm. Math. Phys. 333 1365–1416.
  • [20] Erdős, L. and Knowles, A. (2015). The Altshuler–Shklovskii formulas for random band matrices II: The general case. Ann. Henri Poincaré 16 709–799.
  • [21] Faleiro, E., Gómez, J. M. G., Molina, R. A., Muñoz, L., Relaño, A. and Relamosa, J. (2004). Theoretical derivation of $1/f$ noise in quantum chaos. Phys. Rev. Lett. 93 244101.
  • [22] Fokas, A. S., Its, A. R. and Kitaev, A. V. (1992). The isomonodromy approach to matrix models in $2$D quantum gravity. Comm. Math. Phys. 147 395–430.
  • [23] Fyodorov, Y. V. and Bouchaud, J.-P. (2008). Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 372001, 12.
  • [24] Fyodorov, Y. V., Hiary, G. H. and Keating, J. P. (2012). Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function. Phys. Rev. Lett. 108 170601.
  • [25] Fyodorov, Y. V. and Keating, J. P. (2014). Freezing transitions and extreme values: Random matrix theory, and disordered landscapes. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 20120503, 32.
  • [26] Fyodorov, Y. V., Le Doussal, P. and Rosso, A. (2012). Counting function fluctuations and extreme value threshold in multifractal patterns: The case study of an ideal $1/f$ noise. J. Stat. Phys. 149 898–920.
  • [27] Garoufalidis, S. and Popescu, I. (2013). Analyticity of the planar limit of a matrix model. Ann. Henri Poincaré 14 499–565.
  • [28] Grinblat, L. Š. (1976). A limit theorem for measurable random processes and its applications. Proc. Amer. Math. Soc. 61 371–376.
  • [29] Gustavsson, J. (2005). Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. Henri Poincaré Probab. Stat. 41 151–178.
  • [30] Hughes, C. P., Keating, J. P. and O’Connell, N. (2001). On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220 429–451.
  • [31] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204.
  • [32] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 105–150.
  • [33] Keating, J. P. and Snaith, N. C. (2000). Random matrix theory and $\zeta(1/2+it)$. Comm. Math. Phys. 214 57–89.
  • [34] Krasikov, I. (2011). Some asymptotics for the Bessel functions with an explicit error term. Available at arXiv:1107.2007.
  • [35] Krasovsky, I. V. (2007). Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant. Duke Math. J. 139 581–619.
  • [36] Kuijlaars, A. B. J., McLaughlin, K. T.-R., Van Assche, W. and Vanlessen, M. (2004). The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$. Adv. Math. 188 337–398.
  • [37] Kuijlaars, A. B. J. and Vanlessen, M. (2003). Universality for eigenvalue correlations at the origin of the spectrum. Comm. Math. Phys. 243 163–191.
  • [38] Lytova, A. and Pastur, L. (2009). Central limit theorem for linear eigenvalue statistics of random matrices with independent entries. Ann. Probab. 37 1778–1840.
  • [39] Male, C., Le Caër, G. and Delannay, R. (2007). $1/f^{\alpha}$ noise in the fluctuations of the spectra of tridiagonal random matrices from the $\beta$-Hermite ensemble. Phys. Rev. E 76 042101.
  • [40] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [41] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam.
  • [42] Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., eds. (2010). NIST Handbook of Mathematical Functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC.
  • [43] Pastur, L. and Shcherbina, M. (2011). Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs 171. Amer. Math. Soc., Providence, RI.
  • [44] Rhodes, R. and Vargas, V. (2014). Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 315–392.
  • [45] Rider, B. and Virág, B. (2007). The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN 2 Art. ID rnm006, 33.
  • [46] Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Syst. 1 163–257.
  • [47] Schmitt, F. G. (2003). A causal multifractal stochastic equation and its statistical properties. Eur. Phys. J. B 34 85–98.
  • [48] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [49] Soshnikov, A. (2000). The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 1353–1370.
  • [50] Sosoe, P. and Wong, P. (2013). Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices. Adv. Math. 249 37–87.
  • [51] Szegö, G. (1939). Orthogonal Polynomials. American Mathematical Society Colloquium Publications 23. Amer. Math. Soc., New York.
  • [52] Unterberger, J. (2009). Stochastic calculus for fractional Brownian motion with Hurst exponent $H>\frac{1}{4}$: A rough path method by analytic extension. Ann. Probab. 37 565–614.