Hilbert's irreducibility theorem via random walks

TL;DR

This paper demonstrates that long random walks on certain algebraic groups over number fields rarely hit thin sets, providing new insights into Hilbert's irreducibility theorem and related algebraic structures.

Contribution

It establishes probabilistic bounds for random walks hitting thin sets in algebraic groups over number and function fields, extending Hilbert's irreducibility theorem.

Findings

Random walks rarely hit thin sets in semisimple algebraic groups over number fields.

Results apply to Galois covers, characteristic polynomials, and fixed points in group actions.

Analogous theorems are proved over global function fields.

Abstract

Let $G$ be a connected linear algebraic group over a number field $K$, let $\Gamma$ be a finitely generated Zariski dense subgroup of $G(K)$ and let $Z\subseteq G(K)$ be a thin set, in the sense of Serre. We prove that, if $G/\mathrm{R}_u(G)$ is semisimple and $Z$ satisfies certain necessary conditions, then a long random walk on a Cayley graph of $\Gamma$ hits elements of $Z$ with negligible probability. We deduce corollaries to Galois covers, characteristic polynomials, and fixed points in group actions. We also prove analogous results in the case where $K$ is a global function field.