The norm of products of free random variables

Abstract

Let $X_i$ denote free identically-distributed random variables. This paper investigates how the norm of products $\Pi_n=X_1 X_2 ... X_n$ behaves as $n$ approaches infinity. In addition, for positive $X_i$ it studies the asymptotic behavior of the norm of $Y_n=X_1 \circ X_2 \circ ...\circ X_n$, where $\circ$ denotes the symmetric product of two positive operators: $A \circ B=:A^{1/2}BA^{1/2}$. It is proved that if the expectation of $X_i$ is 1, then the norm of the symmetric product $Y_{n}$ is between $c_1 n^{1/2}$ and $c_2 n$ for certain constant $c_1$ and $c_2$. That is, the growth in the norm is at most linear. For the norm of the usual product $Pi_n$, it is proved that the limit of $n^{-1}\log Norm(Pi_n)$ exists and equals $\log \sqrt{E(X_i^{\ast}X_{i})}.$ In other words, the growth in the norm of the product is exponential and the rate equals the logarithm of the Hilbert-Schmidt norm of operator X. Finally, if $\pi $ is a cyclic representation of the algebra generated by $X_i$, and if $\xi$ is a cyclic vector, then $n^{-1}\log Norm(\pi (\Pi_{n}) \xi)=\log \sqrt{E(X_{i}^{\ast}X_{i})}$ for all $n.$ In other words, the growth in the length of the cyclic vector is exponential and the rate coincides with the rate in the growth of the norm of the product. These results are significantly different from analogous results for commuting random variables and generalize results for random matrices derived by Kesten and Furstenberg.