Irreducibility of random polynomials with large degree

Abstract: I will discuss properties of random polynomials with integer coefficients and large degree. Modulo GRH we show with Peter Varju that they are irreducible modulo cyclotomic factors, thus offering a conditional answer to a question of Odlyzko and Poonen. The proof is based on an understanding of mixing rates of random walks on the affine line over large finite fields and has connections with the Lehmer conjecture.