Mixing time for the asymmetric simple exclusion process in a random environment
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| Format: | Preprint |
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2021
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| _version_ | 1866917592033132544 |
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| author | Lacoin, Hubert Yang, Shangjie |
| author_facet | Lacoin, Hubert Yang, Shangjie |
| contents | We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [ 1, N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-ω_x$ and to site $x+1$ (if $x \le N-1$) at rate $ω_x$ if the target site is not occupied. The sequence $ω=(ω_x)_{ x \in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[ \log ρ_1 ]<0$ where $ρ_1:= (1-ω_1)/ω_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{β+o(1)}$ particles where $β\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}λ, β+ \frac 1 {2λ}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $λ>0$ is such that $\mathbb{E}[ρ_1^λ]=1$ ($λ=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $ω$. We conjecture that our lower bound is sharp up to sub-polynomial correction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2102_02606 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Mixing time for the asymmetric simple exclusion process in a random environment Lacoin, Hubert Yang, Shangjie Probability Mathematical Physics 60K37, 60J27 We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [ 1, N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-ω_x$ and to site $x+1$ (if $x \le N-1$) at rate $ω_x$ if the target site is not occupied. The sequence $ω=(ω_x)_{ x \in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[ \log ρ_1 ]<0$ where $ρ_1:= (1-ω_1)/ω_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{β+o(1)}$ particles where $β\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}λ, β+ \frac 1 {2λ}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $λ>0$ is such that $\mathbb{E}[ρ_1^λ]=1$ ($λ=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $ω$. We conjecture that our lower bound is sharp up to sub-polynomial correction. |
| title | Mixing time for the asymmetric simple exclusion process in a random environment |
| topic | Probability Mathematical Physics 60K37, 60J27 |
| url | https://arxiv.org/abs/2102.02606 |