Random Walks on Homogeneous Spaces by Sparse Solvable Measures

TL;DR

This paper studies the behavior of certain random walks on quotients of SL(k,R), showing they equidistribute and converge exponentially fast to an invariant measure under specific conditions involving measures on diagonal subgroups and smooth curves.

Contribution

It introduces a new class of random walks on homogeneous spaces involving measures supported on smooth curves and diagonal subgroups, proving their equidistribution and exponential convergence.

Findings

Random walks with measures on smooth curves and diagonal subgroups equidistribute.

Convergence to invariant measure occurs exponentially fast.

Results apply to quotients of SL(k,R) with specific measure conditions.

Abstract

The paper analyzes a specific class of random walks on quotients of $X:=\text{SL}(k,{\Bbb R})/ \Gamma$ for a lattice $\Gamma$. Consider a one parameter diagonal subgroup, $\{g_t\}$, with an associated abelian expanding horosphere, $U\cong {\Bbb R}^k$, and let $\phi:[0,1]\rightarrow U$ be a sufficiently smooth curve satisfying the condition that that the derivative of $\phi$ spends $0$ time in any one subspace of ${\Bbb R}^k$. Let $ \mu_U$ be the measure defined as $\phi_*\lambda_{[0,1]},$ where $\lambda_{[0,1]}$ is the Lebesgue measure on $[0,1]$. Let $\mu_A$ be a measure on the full diagonal subgroup of $\text{SL}(k,{\Bbb R})$, such that almost surely the random walk on the diagonal subgroup $A$ with respect to this measure grows exponentially in the direction of the cone expanding $U$. Then the random walk starting at any point $z\in X$, and alternating steps given by $\mu_U$ and $\mu_A$ equidistributes respect to $\text{SL}(k,{\Bbb R})$-invariant measure on $X$. Furthermore, the measure defined by $\mu_A*\mu_U*\dots*\mu_A* \mu_U*\delta_z$ converges exponentially fast to the $\text{SL}(k,{\Bbb R})$-invariant measure on $X$.