Poisson boundary of $GL_d(\Q)$

TL;DR

This paper constructs the Poisson boundary for random walks on the general linear group over rationals, using flag manifolds over p-adic fields and establishing a law of large numbers via Oseledets' theorem.

Contribution

It introduces a novel construction of the Poisson boundary for $GL_d( ext{Q})$ using p-adic flag manifolds and proves a related law of large numbers.

Findings

Poisson boundary characterized by p-adic flag manifolds

Law of large numbers established for the random walk

Framework extends understanding of harmonic analysis on $GL_d( ext{Q})$

Abstract

We construct the Poisson boundary for a random walk supported by the general linear group on the rational numbers as the product of flag manifolds over the $p$-adic fields. To this purpose, we prove a law of large numbers using the Oseledets' multiplicative ergodic theorem.