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Mixing and cut-off for random walks on finite fields and random polynomials

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I will report on joint work with Peter Varj├║ in which we investigate the ax+b random walk on a finite field F_p. Work from the 1990s by Chung-Diaconis-Graham established good upper bounds on mixing time when a=2. We refine their methods to understand the case when a is arbitrary in F_p. Using our previous work on irreducibility of polynomials of large degree, we obtain sharp bounds for the mixing time and prove, conditionally on the Generalized Riemann Hypothesis, that a sharp cut-off occurs.

This talk is part of the Probability series.

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