Hilbert's Irreducibility Theorem via Random Walks.

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/\textrm {R}_{u}(G)$| is either trivial or semisimple and |$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. [ABSTRACT FROM AUTHOR]