论文标题
互换过程中的无限循环在五个维度
Infinite cycles in the interchange process in five dimensions
论文作者
论文摘要
在图$ g =（v，e）$上的交换过程中，将粒子放在$ g $的顶点上，边缘上有独立的泊松时钟。当边缘的时钟时，边缘两侧的两个粒子。这样，随机置换$π_β：v \ to v $在任何时候$β> 0 $。研究的主要对象之一是随机排列的循环结构和长周期的出现。 我们证明了所有维度的$ \ mathbb z ^d $在$ \ mathbb z ^d $上的无限周期的存在。 在我们的证明中，我们研究了一个自我交互的随机步行，称为循环时间随机步行。使用多尺度感应，我们证明它是扩散性的，可以与布朗尼运动结合使用。证明中的关键思想之一是建立当地的逃生财产，该财产表明，当步行以复杂的方式纠缠在其历史上时，步行将很快逃脱。
In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $π_β:V\to V$ is formed for any time $β>0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $β$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.
