论文标题
标记随机连接模型的三角条件
The Triangle Condition for the Marked Random Connection Model
论文作者
论文摘要
我们研究了一个空间随机图模型，其顶点以$ \ mathbb {r}^d $的标记泊松过程给出。根据两个端点的空间位移和它们的标记，将边缘插入任何对点之间，并独立于概率。泊松密度变化后，在轻度条件下发生渗透相变：仅密度有限连接的组件，而对于大密度，几乎可以肯定地存在一个无限的组件。我们的重点是系统至关重要的低密度和高密度阶段之间的过渡。我们证明，如果尺寸足够高并且边缘概率函数满足某些条件，则针对关键连接函数的红外线是有效的。这意味着三角形的条件，因此意味着平均场行为。我们通过将最近确定的泊松过程的蕾丝扩展与光谱估计相结合，从而实现了这一结果。
We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $\mathbb{R}^d$. Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks. Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely. Our focus is on the transition between the low- and high-density phase, where the system is critical. We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid. This implies the triangle condition, and thus mean-field behaviour. We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates.
