论文标题
在高管晶格上的弱限制渗透
Weakly constrained-degree percolation on the hypercubic lattice
论文作者
论文摘要
我们考虑在高立方晶格上的约束度渗透模型，$ \ m athbb l^d =（\ mathbb z^d，\ mathbb e^d）$ for $ d \ geq 3 $。这是由i.i.d. $ [0,1] $的均匀随机变量和正整数（约束）$κ$。 \ Mathbb e^d $中的每个债券$ e \ in Time $ u_e $;它才能成功，并且只有当时它的两个最终媒体属于当时最多属于$κ-1$的开放债券。 我们的主要结果是在关键时间的定量上限，表征了所有$ d \ geq 3 $的相变和$κ$的大多数非平凡值。作为副产品，我们可以获得大限制和尺寸的关键时间是渐近的$ 1/（2D）$。在大多数情况下，以前甚至没有确定相变是非平凡的。 我们证明的成分之一是对临界曲线的改进的上限，$ s _ {\ mathrm {c}}}（b）$，bernoulli混合位点 - 键入在两个维度上，这可能具有独立的兴趣。
We consider the Constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for $d\geq 3$. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in $[0,1]$ and a positive integer (constraint) $κ$. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most $κ-1$ open bonds at that time. Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all $d\geq 3$ and most nontrivial values of $κ$. As a byproduct, we obtain that for large constraints and dimensions the critical time is asymptotically $1/(2d)$. For most cases considered it was previously not even established that the phase transition is nontrivial. One of the ingredients of our proof is an improved upper bound for the critical curve, $s_{\mathrm{c}}(b)$, of the Bernoulli mixed site-bond percolation in two dimensions, which may be of independent interest.
