Cutoff for the transience and mixing time of a SSEP with traps and consequences on the FEP
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910389945499648 |
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| author | Erignoux, Clément Massoulié, Brune |
| author_facet | Erignoux, Clément Massoulié, Brune |
| contents | We introduce a new particle system that we call the SSEP with traps, which is non reversible, attractive, and has a transient regime. We study its \emph{transience time} $θ_K$, meaning the time after which the system is no longer in a transient state with high probability, on the ring with $K$ sites. We first show that $θ_K$ is of order $K^2 \log K$ for a system of size $K$, and more precisely that it exhibits a cutoff at time $\frac{1}{π^2} K^2 \log K$. We then show that its \emph{mixing time} also undergoes cutoff at the same time.
We further define a new mapping between the SSEP with traps and the Facilitated Exclusion Process (FEP) which has attracted significant scrutiny in recent years. We expect that this mapping will be a very useful tool to study the FEP's microscopic and macroscopic behaviour. In particular, using this mapping, we show that the FEP's transience time also undergoes a cutoff at time $\frac{1}{4 π^2} N^2 \log N$. Notably, our results show that for a FEP with particle density strictly greater than $\frac12$, the transient component is exited in a diffusive time. This allows to extend the upper-bound from [Ayre Chleboun 2024] for the mixing time of the FEP with particle density $ρ> 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20010 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cutoff for the transience and mixing time of a SSEP with traps and consequences on the FEP Erignoux, Clément Massoulié, Brune Probability We introduce a new particle system that we call the SSEP with traps, which is non reversible, attractive, and has a transient regime. We study its \emph{transience time} $θ_K$, meaning the time after which the system is no longer in a transient state with high probability, on the ring with $K$ sites. We first show that $θ_K$ is of order $K^2 \log K$ for a system of size $K$, and more precisely that it exhibits a cutoff at time $\frac{1}{π^2} K^2 \log K$. We then show that its \emph{mixing time} also undergoes cutoff at the same time. We further define a new mapping between the SSEP with traps and the Facilitated Exclusion Process (FEP) which has attracted significant scrutiny in recent years. We expect that this mapping will be a very useful tool to study the FEP's microscopic and macroscopic behaviour. In particular, using this mapping, we show that the FEP's transience time also undergoes a cutoff at time $\frac{1}{4 π^2} N^2 \log N$. Notably, our results show that for a FEP with particle density strictly greater than $\frac12$, the transient component is exited in a diffusive time. This allows to extend the upper-bound from [Ayre Chleboun 2024] for the mixing time of the FEP with particle density $ρ> 1/2$. |
| title | Cutoff for the transience and mixing time of a SSEP with traps and consequences on the FEP |
| topic | Probability |
| url | https://arxiv.org/abs/2403.20010 |