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...

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