Mixing of the exclusion process with small bias

We analyze the mixing behavior of the biased exclusion process on a path of length $n$ as the bias $beta_n$ tends to $0$ as $n to infty$. We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when $beta_n$ is of order $1/n$, and the other when $beta_n$ is order $log n/n$.