Mixing Time and Cutoff for the k-SEP

Abstract

We investigate the mixing time of the capacity $k$ simple exclusion process (also called the partial exclusion process) of Schultz and Sandow with $m$ particles on a segment of length $N$. We show that the $k$-SEP exhibits cutoff at time $\frac{1}{2kπ^2}N^2\log m$. 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 $\frac{1}{2kπ^2}N^2\log (kN)$. This extends the celebrated result of Lacoin which determined the mixing time of the symmetric simple exclusion process on a segment of length $N$ and the adjacent transposition shuffle, and proved cutoff in both.

Figures (6)

• Figure 1: One particular configuration of the $k$-SEP with state space $\Omega_{3,6,8}$. The right-pointing arrows have the rates $p_x$ written above them while the left-pointing arrows have $q_x$.
• Figure 2: An example of an element $\sigma \in S_{3,6}$ (i.e. a 3-permutation). For example, $\sigma(3) = \{1,4,7\}$. Blue lines indicate rate 1 swaps in the $S_{k,N}$ shuffle.
• Figure 3: An example showcasing the projection $\varphi$. The top image is an element $\sigma \in S_{2,9}$, the middle is its projection $\varphi_9(\sigma) \in \Omega_{9,2,9}$, and the bottom is the height function $\eta$ associated to $\varphi_9(\sigma)$ (which is also $\tilde{\sigma}(\cdot, 9)$).
• Figure 4: The information obtained from the semi-skeleton associated to the element $\sigma \in S_{3,6}$ given in Figure \ref{['fig:skn-shuffle']} with $R = 3$. In the image, red represents the numbers $\{1,\ldots, 6\}$, blue represents $\{7,\ldots, 12\}$, and green represents $\{13,\ldots,18\}$.
• Figure 5: Example of the action $\sigma\cdot \omega$ with $R=3$. In the image $\omega \in S_{2,3}$ is taken to have $\omega(1) = \{3,5\}, \omega(2) = \{2, 6\}, \omega(3) = \{1,4\}$. Further we take $\sigma = (1\ 2)(3\ 4) \in \tilde{S}$. The top process is the action $\sigma \cdot \omega$ itself while the bottom is the underlying action $\sigma \circ \sigma_\omega$.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3: priestley, lemma 4.1
• Lemma 3.4
• proof : Proof of Theorem \ref{['lower-bd-thm']}
• Lemma 4.1
• proof
• Theorem 4.2: lacoin-at, proposition 3.5