### Investigating the mixing time for the exclusion process in a random environment

Probability Seminar

16th June 2021, 4:00 pm – 5:00 pm

online, online

We consider a system of $k$ particles on a segment of size $N$. Its rules of evolution are the following: a particle at site $x \in \{1 ,\dots, N\}$ jumps to $x+1$ with rate $\omega_x$ and to $x-1$ with rate $1-\omega_x$ where $\omega_x\in (0,1)$, and a jump is cancelled if the site is already occupied. We consider the case where $(\omega_x)^N_{x=1}$ is (the fixed realization of) a sequence of IID random variables. Assuming that $\mathbb E[ \log\frac{1-\omega_x}{\omega_x}]\ne 0$ (that is, transience of the random environment), we prove that this particle systems mixes fast, in the sense that the time that it requires for its distribution to get close to the equilibrium state grows like a power of $N$. We present a lower bound for the power in mixing time which we conjecture to be sharp.

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