Mixing time and cutoff for the k-SPEP

We investigate the mixing time of the capacity $k$ symmetric partial exclusion process of Schütz and Sandow with $m$ particles on a segment of length $N$, and we show that this process exhibits cutoff at time $\frac{1}{2k{\pi }^2}{N}^{2}logm$. We also introduce a related complete multi-species process that we call the $S_{k,N}$ shuffle and show that this process exhibits cutoff at time $\frac{1}{2k{\pi }^2}{N}^{2}log\left(N\right)$. This extends the celebrated result of Lacoin, which proved cutoff for the symmetric simple exclusion process on a segment of length $N$ and the adjacent transposition shuffle.