×

zbMATH — the first resource for mathematics

Lyapunov exponents for some products of random matrices: Exact expressions and asymptotic distributions. (English) Zbl 0584.60018
Random matrices and their applications, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 121-141 (1986).
[For the entire collection see Zbl 0581.00014.]
Consider the Lyapunov exponents \(\mu_ 1^ N\geq...\geq \mu^ N_ N\) for products of i.i.d. \(N\times N\) matrices, A(1),A(2),.... If the \(N^ 2\) entries of A(1) are i.i.d. normal with zero mean and variance 1/N, then for each k, \(\mu^ N_ k=[\log (2/N)+\Psi ((N-k+1)/2)]/2\) where \(\Psi =\Gamma '/\Gamma\) is the digamma function. In this case, the empirical distribution for \(\{\lambda^ N_ k:=\exp (\mu^ N_ k)\}\) converges as \(N\to \infty\) to H(\(\lambda)\), a ”triangle law”, whereas the asymptotic distribution, K, for the (random) eigenvalues of \(| A(1)|:=(A(1)^ TA(1))^{1/2}\) is Wigner’s ”quarter-circle” law.
More generally, when \(| A(1)|\) has a rotationally invariant distribution and a nonrandom limiting K, one can show, under appropriate assumptions, that there is a limiting H(\(\lambda)\) satisfying \[ \int t^ 2[H\lambda^ 2+(1-H)t^ 2]^{-1}dK(t)=1. \] When the entries, \((A(1))_{ij}\), are independent normal with zero mean and variance \(\sigma^ 2_ i/N\) and the \(\sigma_ i's\) have a limiting distribution \(\tilde K,\) one can show, under appropriate assumptions, that there is a limiting H(\(\lambda)\) satisfying \[ \int t^ 2[\lambda^ 2+(1-H)t^ 2]^{-1}d\tilde K(t)=1. \] These identities relating H to K or \(\tilde K\) are expected to be valid under much more general hypotheses on the distribution of A(1) than assumed here.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F99 Limit theorems in probability theory
60G99 Stochastic processes
28D99 Measure-theoretic ergodic theory