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arXiv 2103.17013math.PRmath-phmath.MP
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阶层格点上长程渗流的临界两点函数

The critical two-point function for long-range percolation on the hierarchical lattice

Tom Hutchcroft

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18 pages
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摘要：

我们证明了两个点函数上的常数界（即，点到点连接概率）的临界长程渗流的$d$维层次晶格。更准确地说，我们证明，如果我们连接每对点$x$和$y$的边缘概率$1-\exp(-β\|x-y\|^{-d-α})$，其中$0<α<d$是固定的，$β\geq 0$是一个参数，那么临界两个点函数满足\[ \mathbb{P}_{β_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+α} \]的每对不同的点$x$和$y$。我们推导出特别是该模型具有平均场临界行为时，$α<d/3$和不具有平均场临界行为时，$α>d/3$。

英文摘要：

We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the $d$-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points $x$ and $y$ by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $0<α<d$ is fixed and $β\geq 0$ is a parameter, then the critical two-point function satisfies \[ \mathbb{P}_{β_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+α} \] for every pair of distinct points $x$ and $y$. We deduce in particular that the model has mean-field critical behaviour when $α<d/3$ and does not have mean-field critical behaviour when $α>d/3$.

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