Mixing time of fractional random walk on finite fields

We study a random walk on ${\mathbb{F}}_p$ defined by $X_{n+1}=1\mathrm{/}{X}_n+{\epsilon }_{n+1}$ if $X_n\mathrm{\ne }0$, and $X_{n+1}={\epsilon }_{n+1}$ if $X_n=0$, where ${\epsilon }_{n+1}$ 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 $\log p$, answering a question of Chatterjee and Diaconis.