Equality of Lyapunov and Stability Exponents for Products of Isotropic Random Matrices

Abstract

Abstract. We show that Lyapunov exponents and stability exponents are equal in the case of product of i.i.d isotropic(also known as bi-unitarily invariant) random matrices. We also derive aysmptotic distribution of singular values and eigenvalues of these product random matrices. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of above result, we show probability that product of n i.i.d real isotropic random matrices has all eigenvalues real goes to one as n → ∞. Also, in the proof of a lemma, we observe that a real (complex) Ginibre matrix can be written as product of a random lower triangular matrix and an independent truncated Haar orthogonal (unitary) matrix. Definitions and introductionLet M 1 , M 2 , . . . be sequence of i.i.d random matrices of order d. Define σ n to be diagonal matrix with singular values of product matrix P n = M 1 M 2 ..M n in the diagonal in decreasing order and similarly λ n to be diagonal matrix with eigenvalues of P n in the diagonal in decreasing order of absolute values, for n = 1, 2, . . .. Let |λ n | 1 n and |σ n | 1 n denote diagonal matrices with non-negative n-th roots of absolute values of diagonal entries of λ n and σ n in the diagonal, respectively.Define σ := lim n→∞ |σ n | 1 n and λ := lim n→∞ |λ n | 1 n , if the limits exist. Then diagonal elements of ln σ and ln λ are called Lyapunov exponents and stability exponents for products of i.i.d random matrices, respectively. In other words, they are rates of exponential growth(or decay) of singular values and eigenvalues of product matrices P n , respectively as n → ∞.We consider both real and complex random matrices in this paper.For the sake of simplicity, we restrict ourselves mostly to complex random matrices . But all the definitions and statements, along with proofs, carry over immediately to real case (by replacing everywhere unitary matrices by orthogonal matrices). Definition 1. A random matrix M is said to be isotropic if probability distribution of U M V is same as that of M , for all unitary matrices U, V .They also go by the names of bi-unitarily invariant and rotation invariant random matrices. It follows from definition that distribution of U M V is same as that of M , if U, V are Haar distributed random unitary matrices independent of M and each other. M = P DQ be the singular value decomposition of M , then U M V =

Products of random matrices from polynomial ensembles

“…For completeness, we emphasize that Proposition 5.1 is stated and proved not only for products of bi-unitarily invariant (complex) matrices, but also for products of bi-orthogonally invariant (real) matrices in [55]. Moreover, one can also randomize the order of the singular values and of the eigenvalues of X (M ) , which naturally arises in the alternative derivation of Proposition 5.1 we present below.…”

“…, n. In the past few years, new interest has arisen in this limit due to explicit results for the joint densities of the singular value and the eigenvalues at finite n and M , see [4,24,26,36,40,55]. In particular, the very recent work [55] contains a general result on the Lyapunov and stability exponents which we cite here.…”

“…Proposition 5.1 (Central Limit Theorem for Lyapunov and Stability Exponents [55,Theorem 11]). Let (X j ) j∈N be a sequence of independently and identically distributed, bi-unitarily invariant n × n random matrices such that, with positive probability, X 1 is not a scalar multiple of the identity matrix.…”

“…[4,36] for the induced Laguerre ensemble. Since the components of the vector m are pairwise different [55], both representations, ordered or not, are equivalent and only a matter of taste.…”

“…Interest in this area has resurged more recently due to explicit results for finite products [4,24,26,36,40,55]. In particular, in [55] it is shown that, under certain conditions, for products of independently and identically distributed, bi-unitarily invariant random matrices of fixed dimension, the logarithms of the singular values and the complex eigenvalues are asymptotically Gaussian distributed as the number of factors tends to infinity. We provide a sketch of an alternative proof which is based on the spherical transform and which is reminiscent of the standard Fourier analytical proof of the central limit theorem for sums of random vectors.…”