Cut-off phenomenon for the ax+b Markov chain over a finite field

We study the Markov chain x n + 1 = a x n + b n on a finite field F p , where a ∈ F p × is fixed and b n are independent and identically distributed random variables in F p . Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of a ∈ F p × . We also obtain weaker, but unconditional, upper bounds for the mixing time.