Scattering of products of random matrices

Abstract

Let μ be a probability mesasure on GL(d, ℝ), the multiplicative group of nonsingular d × d matrices, and let {μⁿ} be the sequence of its convolution powers. The aim of this article is to describe certain sufficient conditions for μⁿ(Kx) to converge uniformly (in x) to 0, as n → ∞, for all lompact subsets K of GL(d, ℝ); μ is said to be scattering when this holds. We show in particular that if μ is a probability measure such that the closed subgroup generated by its support, say G, is a noncompact subgroup of SL(d, ℝ) (matrices of determinant 1), then μ is scattering if G acts irreducibly on ℝⁿ or if it is connected; see Corollaries 2.3 and 3.3. The results are essentially contained in [DS1], [DS2] and [JRW]. Our approach here however is elementary, avoiding in particular the theory of Lie groups and algebraic groups on the one hand and intricate probabilistic arguments on the other hand. Also, while the proofs in the above mentioned papers depend crucially on certain results of Csiszar [C] and a theorem proved independently by Mukherjea [M] (see also [HM]) and Derriennic [De] the argument here is self-contained; we start along the first steps as in [C] which we describe in detail and then take a direct route to the desired results. The proofs in [DS1] and [DS2] are refashioned to achieve simplicity. I may also mention here that the present argument also gives an independent proof of the theorem of Mukherjea and Derriennic for matrix groups; see Corollary 3.6.

To Professor K.R. Parthasarathy on his sixtieth birthday