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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2409.16337 |
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| _version_ | 1866914956743540736 |
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| author | Yang, Shangjie |
| author_facet | Yang, Shangjie |
| contents | In this paper, we study the mixing time of the simple exclusion process with $k$ particles in the line segment $[1, N]$ with conductances $c^{(N)}(x, x+1)_{1\le x<N}$ where $c^{(N)}(x, x+1)>0$ is the rate of swapping the contents of the two sites $x$ and $ x+1$. Writing $r^{(N)}(x, x+1) := 1/c^{(N)}(x, x+1)$, under the assumption \begin{equation*}
\limsup_{N\to \infty}\, \frac{1}{N}\sup_{1< m \le N}\, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|\;=\;0\,, \end{equation*} and some further assumptions on $r^{(N)}(x, x+1)_{x \in \mathbb{N} }$ and $k$, we prove that around time $(1+o(1)) (2 π^2)^{-1} N^2 \log k$, the total variation distance to equilibrium of the simple exclusion process drops abruptly from $1$ to $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_16337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cutoff of the simple exclusion process with inhomogeneous conductances Yang, Shangjie Probability In this paper, we study the mixing time of the simple exclusion process with $k$ particles in the line segment $[1, N]$ with conductances $c^{(N)}(x, x+1)_{1\le x<N}$ where $c^{(N)}(x, x+1)>0$ is the rate of swapping the contents of the two sites $x$ and $ x+1$. Writing $r^{(N)}(x, x+1) := 1/c^{(N)}(x, x+1)$, under the assumption \begin{equation*} \limsup_{N\to \infty}\, \frac{1}{N}\sup_{1< m \le N}\, \left| \sum_{x=2}^m r^{(N)}(x-1, x)- (m-1) \right|\;=\;0\,, \end{equation*} and some further assumptions on $r^{(N)}(x, x+1)_{x \in \mathbb{N} }$ and $k$, we prove that around time $(1+o(1)) (2 π^2)^{-1} N^2 \log k$, the total variation distance to equilibrium of the simple exclusion process drops abruptly from $1$ to $0$. |
| title | Cutoff of the simple exclusion process with inhomogeneous conductances |
| topic | Probability |
| url | https://arxiv.org/abs/2409.16337 |