Random walks on linear groups satisfying a Schubert condition
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RANDOM WALKS ON LINEAR GROUPS SATISFYING A SCHUBERT CONDITION BY
Weikun He∗ Einstein Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel e-mail: [email protected]
ABSTRACT
We study random walks on GLd (R) whose proximal dimension r is larger than 1 and whose limit set in the Grassmannian Grr,d (R) is not contained any Schubert variety. These random walks, without being proximal, behave in many ways like proximal ones. Among other results, we establish a H¨ older-type regularity for the stationary measure on the Grassmannian associated to these random walks. Using this and a generalization of Bourgain’s discretized projection theorem, we prove that the proximality assumption in the Bourgain–Furman–Lindenstrauss–Mozes theorem can be relaxed to this Schubert condition.
1. Introduction Let d ≥ 2 and let μ be a Borel probability measure on GLd (R). Let Γμ denote the closed subsemigroup generated by the support of μ. The random walk on GLd (R) associated to μ is (gn · · · g1 )n≥0 where (gn )n≥1 is a sequence of independent and identically distributed random variables distributed according to μ. Thus, the distribution of the random walk at time n ≥ 0 is μ∗n , the multiplicative convolution of μ with itself n times. The study of asymptotic behaviors of these random walks, known as the theory of random matrix products, dates back to the 60’s. In this theory, a condition called proximality (also known as contraction) plays an important ∗ The author is supported by ERC grant ErgComNum 682150.
Received April 10, 2019 and in revised form July 21, 2019
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role. In this article, we define a property that can be seen as a weak version of the proximality. The aim is then to find and prove, under this weaker assumption, results analogous to those which are already known under the proximality assumption. Let us start by defining this property which we will call (S) in this article. In order to do so, recall some definition. The proximal dimension of a subsemigroup Γ ⊂ GLd (R) is rΓ = min{rk π | π ∈ RΓ \ {0}}, where RΓ denotes the closure in End(Rd ) of the set of all elements of the form λg with λ ∈ R and g ∈ Γ. Since this notion of proximal dimension depends on the specific embedding of Γ into some GLd , it is better to refer to this quantity rΓ as the proximal dimension of the representation Rd of Γ or as the proximal dimension of the action of Γ on Rd . Thus, Γ is proximal if and only if its proximal dimension is equal to 1. We define ΠΓ = {π ∈ RΓ | rk π = rΓ }. Let Gr(rΓ , d) denote the Grassmannian of rΓ -dimensional linear subspaces of Rd . The limit set of Γ in the Grassmannian is defined as LΓ = {im π ∈ Gr(rΓ , d) | π ∈ ΠΓ }. Definition 1.1: We say that Γ has property (S) if its limit set in Gr(rΓ , d) is not contained in any proper Schubert variety. Equivalently, (S)
∀W ∈ Gr(d − rΓ , d), ∃V ∈ LΓ , V ∩ W = {0}.
For example, if the action of Γ on Rd is irreducible and proximal then (S) is automatically satisfied. Let G denote the Zariski closure of Γ in
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