Abstract
A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the explicit form of the distribution of the determinant. We use these result to evaluate the sum of the Lyapunov spectrum of the corresponding random matrix product, and we further give explicit expressions for the largest Lyapunov exponent. Generalisations to the case of complex or quaternion entries are also given. For standard Gaussian matrices X, the full Lyapunov spectrum for products of random matrices \(I_N + {1 \over c} X\) is computed in terms of a generalised hypergeometric function in general, and in terms of a single single integral involving a modified Bessel function for the largest Lyapunov exponent.
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References
Akemann, G., Burda, Z., Kieburg, M.: Universal distribution of Lyapunov exponents for products of Ginibre matrices. J. Phys. A 47, 395202 (2014)
Bartlett, M.S.: The vector representation of a sample. Math. Proc. Camb. Philos. Soc. 30, 327–340 (1934)
Bausch, J.: On the efficient calculation of a linear combination of chi-squared random variables with an application in counting string vacua. J. Phys. A 46, 505202 (2013)
Cohen, J.E., Newman, C.M.: The stability of large random matrices and their products. Ann. Prob. 12, 283–310 (1984)
Constantine, A.G.: Some noncentral distribution problems in multivariate analysis. Ann. Math. Stat. 34, 1270–1285 (1963)
Constantine, A.G., Muirhead, R.J.: Asymptotic expansions for distributions of latent roots in multivariate analysis. J. Mult. Anal. 6, 369–391 (1976)
Cui, X., Zhang, Q.T.: Generic procedure for tightly bounding the capacity of MIMO correlated Rician fading channels. IEEE Trans. Commun. 53, 890–898 (2005)
Díaz-García, J.A., Gutiérrez-Jáimez, R.: Random matrix theory and multivariate statistics. arXiv:0907.1064
Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)
Forrester, P.J.: Lyapunov exponents for products of complex Gaussian random matrices. J. Stat. Phys. 151, 796–808 (2013)
Forrester, P.J.: Asymptotics of finite system Lyapunov exponents for some random matrix ensembles. J. Phys. A 48, 215205 (2015)
Forrester, P.J.: Matrix polar decomposition and generalisations of the Blaschke–Petkantschin formula in integral geometry. arXiv:1701.04505
Forrester, P.J.: Comment on “Sum of squares of uniform random variables” by I. Weissman. Stat. Probab. Lett. 142, 118–122 (2018)
Forrester, P.J., Ipsen, J.R.: Selberg integral theory and Muttalib–Borodin ensembles. Adv. Appl. Math. 95, 152–176 (2018)
Forrester, P.J., Zhang, J.: Volumes and distributions for random unimodular complex and quaternion lattices. J. Number Theory (2018). https://doi.org/10.1016/j.jnt.2018.03.010
Ipsen, J.R.: Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices. J. Phys. A 48, 155204 (2015)
Kabluchko, Z., Temesvari, D., Thäle, C.: Expected intrinsic volumes and facet number of random beta-polytopes. arXiv:1707.02253
Kabluchko, Z., Marynych, A., Temesvari, D., Thäle, C.: Cones generated by random points on half-spheres and convex hulls of Poisson point processes. arXiv:1801.08008
Kargin, V.: On the largest Lyapunov exponent for products of Gaussian matrices. J. Stat. Phys. 157, 70–83 (2014)
Mathai, A.M.: Random \(p\)-content of a \(p\)-parallelotope in Euclidean \(n\)-space. Adv. Appl. Probab. 31, 343–354 (1999)
Mathai, A.M.: An Introduction to Geometric Probability. Gordon and Breach Science Publishers, Amsterdam (1999)
Moghadasi, S.R.: Polar decomposition of the \(k\)-fold Lebesgue measure on \({\mathbb{R}}^n\). Bull. Aust. Math. Soc. 85, 315–324 (2012)
Muirhead, R.J.: Latent roots and matrix variates: a review of some asymptotic results. Ann. Stat. 6, 5–33 (1978)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Newman, C.M.: The distribution of Lyapunov exponents: exact results for random matrices. Commun. Math. Phys. 103, 121–126 (1986)
Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968)
Paris, R.B.: Exponentially small expansions of the confluent hypergeometric function. Appl. Math. Sci. 7, 6601–6609 (2013)
Prékopa, A.: On random determinants I. Studia Sci. Math. Hung. 2, 125–132 (1967)
Raghunathan, M.S.: A proof of Oseledec’s multiplicative ergodic theorem. Israel J. Math. 32, 356–362 (1979)
Rouault, A.: Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles. ALEA 3, 181–230 (2007)
Ruben, H.: The volume of an isotropic random parallelotope. J. Appl. Probab. 16, 84–94 (1979)
Sim, C.H.: Point processes with correlated gamma inter arrival times. Stat. Probab. Lett. 15, 135–141 (1992)
Tsaig, Y., Donoho, D.L.: Breakdown of equivalence between the minimal \(L_1\)-norm solution and the sparsest solution. Signal Process. 86, 533–548 (2006)
Weissman, I.: Sum of squares of uniform random variables. Stat. Probab. Lett. 129, 147–154 (2017)
Wikipedia, Noncentral chi-squared distribution. https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
Zanon, N., Derrida, B.: Weak disorder expansion of Liapunov exponents in a degenerate case. J. Stat. Phys. 50, 509–528 (1988)
Acknowledgements
This research project is part of the program of study supported by the ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS), and the Australian Research Council Discovery Project Grant DP170102028. The work of JZ was supported by a University of Melbourne Research Scholarship, and an ACEMS top-up scholarship.
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Communicated by Ivan Corwin.
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Forrester, P.J., Zhang, J. Lyapunov Exponents for Some Isotropic Random Matrix Ensembles. J Stat Phys 180, 558–575 (2020). https://doi.org/10.1007/s10955-019-02474-2
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DOI: https://doi.org/10.1007/s10955-019-02474-2