From quenched invariance principle to semigroup convergence with applications to exclusion processes
From quenched invariance principle to semigroup convergence with applications to exclusion processes
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发表时间：
2023-03
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通讯作者：
Alberto Chiarini;Simone Floreani;Federico Sau
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作者：
Alberto Chiarini;Simone Floreani;Federico Sau
考虑在平移不变和遍历随机环境中从原点开始在 $\mathbb{Z}^d$ 上进行随机游走。在这篇简短的文章中，假设适当重新调整游走的淬灭不变性原理成立，我们将展示如何导出相应半群的 $L^1$ 收敛。然后，我们应用这个结果来获得 $\mathbb{Z}^d$、$d\ge 2$ 上简单对称排除过程的猝灭路径流体动力学极限，其中 i.i.d.对称最近邻电导 $\omega_{xy}\in [0,\infty)$ 仅满足 $$\mathbb{Q}(\omega_{xy}>0)>p_c\ ,$$ 其中 $p_c$ 是键渗透的临界值。
Consider a random walk on $\mathbb{Z}^d$ in a translation-invariant and ergodic random environment and starting from the origin. In this short note, assuming that a quenched invariance principle for the opportunely-rescaled walks holds, we show how to derive an $L^1$-convergence of the corresponding semigroups. We then apply this result to obtain a quenched pathwise hydrodynamic limit for the simple symmetric exclusion process on $\mathbb{Z}^d$, $d\ge 2$, with i.i.d. symmetric nearest-neighbors conductances $\omega_{xy}\in [0,\infty)$ only satisfying $$\mathbb{Q}(\omega_{xy}>0)>p_c\ ,$$ where $p_c$ is the critical value for bond percolation.