论文标题
从度量图和离散GFF的跨越估计值
Crossing estimates from metric graph and discrete GFF
论文作者
论文摘要
We compare level-set percolation for Gaussian free fields (GFFs) defined on a rectangular subset of $δ\mathbb{Z}^2$ to level-set percolation for GFFs defined on the corresponding metric graph as the mesh size $δ$ goes to 0. In particular, we look at the probability that there is a path that crosses the rectangle in the horizontal direction on which the field is positive.我们在离散图中表明，这种概率严格较大。在公制图案中，我们表明，对于适当的边界条件，水平交叉事件的闭合枢轴边缘的概率在$Δ$中对数衰减。在离散的图形情况下，我们计算适当边界条件的水平交叉概率的极限。
We compare level-set percolation for Gaussian free fields (GFFs) defined on a rectangular subset of $δ\mathbb{Z}^2$ to level-set percolation for GFFs defined on the corresponding metric graph as the mesh size $δ$ goes to 0. In particular, we look at the probability that there is a path that crosses the rectangle in the horizontal direction on which the field is positive. We show this probability is strictly larger in the discrete graph. In the metric graph case, we show that for appropriate boundary conditions the probability that there exists a closed pivotal edge for the horizontal crossing event decays logarithmically in $δ$. In the discrete graph case, we compute the limit of the probability of a horizontal crossing for appropriate boundary conditions.
