论文标题
激活随机步行的分离截止
Separation cutoff for Activated Random Walks
论文作者
论文摘要
我们考虑在任意有限网络上激活的随机步行，粒子被随机插入并在边界上吸收。尽管动力学和缺乏对固定分布的知识不可逆，但我们明确确定了过程的放松时间，并证明分离截止值等于产品条件。我们还对截止窗口的中心和宽度进行了尖锐的估计。最后，我们通过在各个网络上建立明确的分离截止，包括：（i）任何固定的无限无可能的不可符号的大量有限亚图，在边界处具有吸收和（ii）（ii）在单个顶端吸收的大量有限顶点传播图。后者的结果解决了莱文和梁的猜想。我们的证明依赖于对莱文和梁最近发现的强大固定时间的精致分析，并涉及IDLA过程。
We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fixed infinite non-amenable graph, with absorption at the boundary and (ii) large finite vertex-transitive graphs with absorption at a single vertex. The latter result settles a conjecture of Levine and Liang. Our proofs rely on the refined analysis of a strong stationary time recently discovered by Levine and Liang and involving the IDLA process.
