The Annals of Probability

Random matrices: Law of the determinant

Hoi H. Nguyen and Van Vu

Full-text: Open access

Abstract

Let $A_{n}$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_{n}|$ satisfies a central limit theorem. More precisely,

\begin{eqnarray*}&&\sup_{x\in{\mathbf {R}}}\biggl|{\mathbf{P} }\biggl(\frac{\log(|\det A_{n}|)-({1}/{2})\log(n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf{P} }\bigl(\mathbf{N} (0,1)\le x\bigr)\biggr|\\&&\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}

Article information

Source
Ann. Probab., Volume 42, Number 1 (2014), 146-167.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1389278522

Digital Object Identifier
doi:10.1214/12-AOP791

Mathematical Reviews number (MathSciNet)
MR3161483

Zentralblatt MATH identifier
1299.60005

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F05: Central limit and other weak theorems

Keywords
Random matrices random determinant

Citation

Nguyen, Hoi H.; Vu, Van. Random matrices: Law of the determinant. Ann. Probab. 42 (2014), no. 1, 146--167. doi:10.1214/12-AOP791. https://projecteuclid.org/euclid.aop/1389278522


Export citation

References

  • [1] Bai, Z. and Silverstein, J. (2006). Spectral Analysis of Large Dimensional Random Matrices. Science press, Beijing.
  • [2] Berry, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49 122–136.
  • [3] Bourgain, J., Vu, V. H. and Wood, P. M. (2010). On the singularity probability of discrete random matrices. J. Funct. Anal. 258 559–603.
  • [4] Dembo, A. (1989). On random determinants. Quart. Appl. Math. 47 185–195.
  • [5] El Machkouri, M. and Ouchti, L. (2007). Exact convergence rates in the central limit theorem for a class of martingales. Bernoulli 13 981–999.
  • [6] Erdős, L. Universality of Wigner random matrices: A survey of recent results. Available at arXiv:1004.0861v2.
  • [7] Forsythe, G. E. and Tukey, J. W. (1952). The extent of n random unit vectors. Bull. Amer. Math. Sot. 58 502.
  • [8] Gīrko, V. L. (1979). A central limit theorem for random determinants. Theory Probab. Appl. 24 729–740.
  • [9] Girko, V. L. (1990). Theory of Random Determinants. Mathematics and Its Applications (Soviet Series) 45. Kluwer Academic, Dordrecht. Translated from the Russian.
  • [10] Girko, V. L. (1997). A refinement of the central limit theorem for random determinants. Theory Probab. Appl. 42 121–129.
  • [11] Goodman, N. R. (1963). The distribution of the determinant of a complex Wishart distributed matrix. Ann. Math. Statist. 34 178–180.
  • [12] Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 5 119–136 (electronic).
  • [13] Kahn, J., Komlós, J. and Szemerédi, E. (1995). On the probability that a random $\pm1$-matrix is singular. J. Amer. Math. Soc. 8 223–240.
  • [14] Komlós, J. (1967). On the determinant of $(0,1)$ matrices. Studia Sci. Math. Hungar. 2 7–21.
  • [15] Komlós, J. (1968). On the determinant of random matrices. Studia Sci. Math. Hungar. 3 387–399.
  • [16] Nyquist, H., Rice, S. O. and Riordan, J. (1954). The distribution of random determinants. Quart. Appl. Math. 12 97–104.
  • [17] Prékopa, A. (1967). On random determinants. I. Studia Sci. Math. Hungar. 2 125–132.
  • [18] Rempała, G. and Wesołowski, J. (2005). Asymptotics for products of independent sums with an application to Wishart determinants. Statist. Probab. Lett. 74 129–138.
  • [19] Rouault, A. (2007). Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles. ALEA Lat. Am. J. Probab. Math. Stat. 3 181–230.
  • [20] Rudelson, M. and Vershynin, R. (2008). The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218 600–633.
  • [21] Szekered, G. and Turán, P. (1937). On an extremal problem in the theory of determinants (Hungarian). Math. Naturwiss. Am. Ungar. Akad. Wiss. 56 796–806.
  • [22] Tao, T. and Vu, V. Random matrices: The universality phenomenon for Wigner ensembles. Available at arXiv:1202.0068v1.
  • [23] Tao, T. and Vu, V. (2006). On random $\pm1$ matrices: Singularity and determinant. Random Structures Algorithms 28 1–23.
  • [24] Tao, T. and Vu, V. (2007). On the singularity probability of random Bernoulli matrices. J. Amer. Math. Soc. 20 603–628.
  • [25] Tao, T. and Vu, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261–307.
  • [26] Tao, T. and Vu, V. (2010). Random matrices: The distribution of the smallest singular values. Geom. Funct. Anal. 20 260–297.
  • [27] Tao, T. and Vu, V. (2010). Smooth analysis of the condition number and the least singular value. Math. Comp. 79 2333–2352.
  • [28] Tao, T. and Vu, V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127–204.
  • [29] Turán, P. (1955). On a problem in the theory of determinants. Acta Math. Sinica 5 411–423.