Mixing time of fractional random walk on finite fields

We study a random walk on $mathbb{F}_p$ defined by $X_{n+1}=1/X_n+varepsilon_{n+1}$ if $X_n eq 0$, and $X_{n+1}=varepsilon_{n+1}$ if $X_n=0$, where $varepsilon_{n+1}$ 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.