The norm of products of free random variables

Abstract

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: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 P in, it is proved that the limit of n −1 log N orm(P in) 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 −1 log N orm(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. IntroductionSuppose X 1 , X 2 , ... , X n are identically-distributed free random variables. These variables are infinite-dimensional linear operators but the reader may find it convenient to think of them as very large random matrices. The first question we will address in this paper is how the norm of Π n = X 1 X 2 ...X n behaves. If X i are all positive, then it is natural to look also at the symmetric product operation • defined as follows:The benefit is that unlike the usual operator product, this operation maps the set of positive variables to itself. For this operation we can ask how the norm of symmetric products Y n = X 1 • X 2 • ... • X n behaves. 1 * Courant Institute of Mathematical Sciences; 109-20 71st Road, Apt. 4A, Forest Hills NY 11375; kargin@cims.nyu.edu1 The operation • is neither commutative, nor associative. By convention we multiply starting on the right, so, for example, X 1 • X 2 • X 3 • X 4 = X 1 • (X 2 • (X 3 • X 4 )) . However,

Products of free random variables and $k$-divisible non-crossing partitions

“…Our main result can be used to compute bounds for the supports of free multiplicative convolutions of positive measures. Our results in this section generalize those by Kargin in [5], where the case µ 1 = · · · = µ k was treated. We recall that the cardinality of N C(n) is given by the Catalan number C n , which is easily bounded by 4 n .…”

“…. , a k ), (5) where N C k (n) and N C k (n) denote, respectively, the k-equal and k-divisible partitions of [kn].…”

On the Largest Lyapunov Exponent for Products of Gaussian Matrices

“…The result of Theorem 1.2 is in agreement with the free probability prediction (see, for example, [9]). FIGURE 1.…”

An Introduction to Random Matrices

“…It was also shown in [GuS08] to hold with matrices whose laws are absolutely continuous with respect to the Lebesgue measure and possess a strictly log-concave density. The norm of long words in free non-commutative variables is discussed in [Kar07a]. We note that a byproduct of the proof of Theorem 5.5.1 is that the Stieltjes transform of the law of any self-adjoint polynomial in free semicircular random variables is an algebraic function, as one sees by applying the algebraicity criterion [AnZ08b, Theorem 6.1], to the Schwinger-Dyson equation as expressed in the form (5.5.32).…”