Hilbert's Irreducibility Theorem via Random Walks

Let be a connected linear algebraic group over a number field, let be a finitely generated Zariski dense subgroup of, and let be a thin set, in the sense of Serre. We prove that, if is either trivial or semisimple and satisfies certain necessary conditions, then a long random walk on a Cayley graph of hits elements of 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 is a global function field.