Lyapunov exponents and eigenvalues of products of random matrices

Abstract: Let $X_1,X_2, \ldots $ be a sequence of $i.i.d$ real (complex) $d \times d $ invertible random matrices with common distribution $\mu$ and $\sigma_1(n), \sigma_2(n), \ldots , \sigma_d(n)$ be the singular values, $\lambda_1(n), \lambda_2(n), \ldots , \lambda_d(n)$ be the eigenvalues of $X_nX_{n-1}\cdots X_1$ in the decreasing order of their absolute values for every $n$. It is known that if $\mathbb{E}(\log^{+}|X_1|)< \infty$, then with probability one for all $1 \leq p \leq d$, $$ \lim_{n \to \infty} \frac{1}{n}\log \sigma_p(n)=\gamma_p, $$ where ${\gamma_1,\gamma_2 \ldots \gamma_d}$ are the Lyapunov exponents associated with $\mu$. In this paper we show that under certain support and moment conditions on $\mu$, the absolute values of eigenvalues also exhibit the same asymptotic behaviour. In fact, a stronger asymptotic relation holds between the singular values and the eigenvalues $i.e.$ for any $r>0$ with probability one for all $1 \leq p \leq d$, $$ \lim_{n \to \infty} \frac{1}{n^r}\log \left(\frac{|\lambda_p(n)|}{\sigma_p(n)}\right)= 0, $$ which implies that the fluctuations of the eigenvalues have the same asymptotic distribution as that of the corresponding singular values. Isotropic random matrices and also random matrices with $i.i.d$ real elements, which have some finite moment and bounded density whose support contains an open set, are shown to satisfy the moment and support conditions under which the above relations hold.