Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912249396854784 |
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| author | Conroy, Michael Sethuraman, Sunder |
| author_facet | Conroy, Michael Sethuraman, Sunder |
| contents | We consider the symmetric simple exclusion system on $\mathbb{Z}^d$, $d \ge 2$, starting from a class of ``step'' initial conditions in which particles are constrained within a half-space. One may count the number $N_t$ of particles that have moved beyond a distance $z = z(t)$ into the initially-empty half of $\mathbb{Z}^d$ at time $t$. We show in large generality that when $\lim_{t\to\infty} E[N_t]$ exists, correlations between particles beyond $z$ vanish as $t \to \infty$ so as to allow convergence of $N_t$ to the same Poisson distribution one would get were the particles allowed to move independently. When the initial condition constrains a region of polynomial growth, we identify $z(t)$ and the limit of $E[N_t]$ explicitly. As a consequence of the limit, we obtain a Gumbel limit distribution for the extremal particle position, as well as the limiting distributions of all order statistics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10522 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1 Conroy, Michael Sethuraman, Sunder Probability 60K35, 60F05 We consider the symmetric simple exclusion system on $\mathbb{Z}^d$, $d \ge 2$, starting from a class of ``step'' initial conditions in which particles are constrained within a half-space. One may count the number $N_t$ of particles that have moved beyond a distance $z = z(t)$ into the initially-empty half of $\mathbb{Z}^d$ at time $t$. We show in large generality that when $\lim_{t\to\infty} E[N_t]$ exists, correlations between particles beyond $z$ vanish as $t \to \infty$ so as to allow convergence of $N_t$ to the same Poisson distribution one would get were the particles allowed to move independently. When the initial condition constrains a region of polynomial growth, we identify $z(t)$ and the limit of $E[N_t]$ explicitly. As a consequence of the limit, we obtain a Gumbel limit distribution for the extremal particle position, as well as the limiting distributions of all order statistics. |
| title | Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1 |
| topic | Probability 60K35, 60F05 |
| url | https://arxiv.org/abs/2501.10522 |