Hilbert's Irreducibility Theorem via Random Walks.

The article focuses on quantitative generalizations of Hilbert's irreducibility theorem (HIT) for connected linear algebraic groups over global fields, particularly analyzing the probability that a long random walk on a finitely generated Zariski dense subgroup hits a thin subset, as defined by Serre. It establishes that if the quotient of the group by its unipotent radical is trivial or semisimple and the thin set satisfies certain ramification conditions, then the probability that the random walk lands in the thin set decays to zero, exponentially in the semisimple case and polynomially otherwise. Applications include results on the irreducibility of fibers in covers, typical Galois groups of characteristic polynomials associated with group elements, and the rarity of elements fixing rational points under group actions. The paper also extends these results to global function fields under suitable assumptions, employing strong and superstrong approximation theorems and the group large sieve method to obtain these probabilistic bounds.