Mixing time of fractional random walk on finite fields

Abstract

We study a random walk on ${\mathbb{F}}_p$ defined by $X_{n+1}=1∕X_n+{\mathit{\epsilon }}_{n+1}$ if $X_n\ne 0$, and $X_{n+1}={\mathit{\epsilon }}_{n+1}$ if $X_n=0$, where ${\mathit{\epsilon }}_{n+1}$ are independent and identically distributed. This can be seen as a non-linear analogue of the Chung–Diaconis–Graham process. We show that the mixing time is of order $logp$, answering a question of Chatterjee and Diaconis.