论文标题
正方形的渗透和随机右角的二次发散的阈值
Square percolation and the threshold for quadratic divergence in random right-angled Coxeter groups
论文作者
论文摘要
给定图$γ$，其辅助\ emph {square-graph} $ \ square（γ）$是其顶点的图形，其顶点是$γ$的非边缘，其边缘是$γ$中诱导平方的非边缘（即$ 4 $ cycecle）。我们确定阈值边缘验证性$ p = p_c（n）$，其中Erd {\ h o} s--rényi随机图$γ=γ_{n，p} $几乎肯定地肯定具有与$γ_{n，p} $ umgγ_{n，p}的正方形组合在一起的连接组件的正方形图。我们显示$ p_c（n）= \ sqrt {\ sqrt {\ sqrt {6} -2}/\ sqrt {n} $，$ p_c（n）$的早期范围的polyrogarithmic改进。作为推论，我们确定阈值$ p = p_c（n）$，随机右角coxeter组$ w_ {γ_{γ_{n，p}} $渐近地肯定几乎肯定会变得强烈地代数$ 1 $ $ 1 $，并且具有Quadratic Divergence。
Given a graph $Γ$, its auxiliary \emph{square-graph} $\square(Γ)$ is the graph whose vertices are the non-edges of $Γ$ and whose edges are the pairs of non-edges which induce a square (i.e., a $4$-cycle) in $Γ$. We determine the threshold edge-probability $p=p_c(n)$ at which the Erd{\H o}s--Rényi random graph $Γ=Γ_{n,p}$ begins to asymptotically almost surely have a square-graph with a connected component whose squares together cover all the vertices of $Γ_{n,p}$. We show $p_c(n)=\sqrt{\sqrt{6}-2}/\sqrt{n}$, a polylogarithmic improvement on earlier bounds on $p_c(n)$ due to Hagen and the authors. As a corollary, we determine the threshold $p=p_c(n)$ at which the random right-angled Coxeter group $W_{Γ_{n,p}}$ asymptotically almost surely becomes strongly algebraically thick of order $1$ and has quadratic divergence.
