Abstract In the Constrained-degree percolation model on a graph ( V , E ) there are a sequence, ( U e ) e ∈ E , of i.i.d. random variables with distribution U [ 0 , 1 ] and a positive integer k . Each bond e tries to open at time U e , it succeeds if both its end-vertices would have degrees at most k − 1 . We prove a phase transition theorem for this model on the square lattice L 2 , as well as on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase.

中文翻译：

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约束度渗透模型

摘要 在图 ( V , E ) 上的约束度渗透模型中，有一个序列 (U e ) e ∈ E ，由分布为 U [ 0 , 1 ] 的 iid 随机变量和一个正整数 k 组成。每个债券 e 在时间 U e 尝试打开，如果它的两个末端顶点的度数最多为 k − 1 ，它就会成功。我们在方格 L 2 以及 d 元正则树上证明了该模型的相变定理。我们还证明了在方格上无限簇在超临界相中是唯一的。