Interpolation percolation

Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^2$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.

布朗最后通道渗流中半无穷测地线与竞争界面的全局结构

Global structure of semi-infinite geodesics and competition interfaces in Brownian last-passage percolation

本文研究布朗最后通道渗流中半无穷测地线与竞争界面的全局结构，通过推导所有方向上 Busemann 函数的联合分布，揭示了不可数初始点引发的新现象，并刻画了测地线分叉、合并及竞争界面非平凡点集的 Hausdorff 维数等性质。

Scale-free percolation

We formulate and study a model for inhomogeneous long-range percolation on $\Zbold^d$. Each vertex $x\in\Zbold^d$ is assigned a non-negative weight $W_x$, where $(W_x)_{x\in\Zbold^d}$ are i.i.d.\ random variables. Conditionally on the weights, and given two parameters $α,λ>0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{xy}=1-\exp\{-λW_xW_y/|x-y|^α\}$. The parameter $λ$ is the percolation parameter, while $α$ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exis…