论文标题
珠模型的连续Kasteleyn理论
Continuous Kasteleyn theory for the bead model
论文作者
论文摘要
考虑半混凝土圆环$ \ mathbb {t} _n：= [0,1）\ times \ {0,1，\ ldots，n-1 \} $代表$ n $ n $单位长度字符串并行运行。 $ \ mathbb {t} _n $上的珠子配置是$ \ mathbb {t} _n $上的一个点过程，该属性是同一字符串上每两个连续点之间的属性，每个相邻字符串都在一个点上。在本文中，我们开发了Kasteleyn理论的连续版本，以表明在$ \ Mathbb {T} _n $上的珠子配置功能可以根据$ \ Mathbb {t} _n $上的某些操作员的弗雷姆决定因素表示。我们获得了$ \ mathbb {t} _n $上珠构型卷的显式公式。该公式的渐近学证实了自由概率文献中的最新预测。此后，我们在$ \ mathbb {t} _n $上研究随机珠构型，表明它们具有可以与排除过程连接的确定结构。我们使用此机械来构建环上Tasep的新概率表示。
Consider the semi-discrete torus $\mathbb{T}_n := [0,1) \times \{0,1,\ldots,n-1\}$ representing $n$ unit length strings running in parallel. A bead configuration on $\mathbb{T}_n$ is a point process on $\mathbb{T}_n$ with the property that between every two consecutive points on the same string, there lies a point on each of the neighbouring strings. In this article we develop a continuous version of Kasteleyn theory to show that partition functions for bead configurations on $\mathbb{T}_n$ may be expressed in terms of Fredholm determinants of certain operators on $\mathbb{T}_n$. We obtain an explicit formula for the volumes of bead configurations on $\mathbb{T}_n$. The asymptotics of this formula confirm a recent prediction in the free probability literature. Thereafter we study random bead configurations on $\mathbb{T}_n$, showing that they have a determinantal structure which can be connected with exclusion processes. We use this machinery to construct a new probabilistic representation of TASEP on the ring.
