The norm of products of free random variables

Let Xi denote free identically-distributed random variables. This paper investigates how the norm of products Πn = X1X2...Xn behaves as n approaches infinity. In addition, for positive Xi it studies the asymptotic behavior of the norm of Yn = X1 ◦ X2 ◦ ... ◦ Xn, where ◦ denotes the symmetric product of two positive operators: A ◦B =: ABA. It is proved that if the expectation of Xi is 1, then the norm of the symmetric product Yn is between c1n 1/2 and c2n for certain constant c1 and c2. That is, the growth in the norm is at most linear. For the norm of the usual product Pin, it is proved that the limit of n logNorm(Pin) exists and equals log p E (X∗ i Xi). 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 π is a cyclic representation of the algebra generated by Xi, and if ξ is a cyclic vector, then n logNorm(π (Πn) ξ) = log p E (X∗ i Xi) 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.