论文标题
在$ \ mathbb {z}^d $上的远程渗透的关键两点函数上的尖锐层次上限
Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on $\mathbb{Z}^d$
论文作者
论文摘要
考虑在$ \ mathbb {z}^d $上进行远距离的bernoulli渗透，其中我们将每对不同点$ x $和$ y $连接到边缘，概率$ 1- \ exp（-β\ | x-y \ |^|^|^{ - d-α}）$，其中$α> 0 $ nesive和$β\ egeq^egeq 0 a parameter。我们证明，如果$ 0 <α<d $，那么关键的两点函数满足\ [\ frac {1} {|λ_r|} \ sum_ {x \inλ_r} \ c \inλ_r} \ mathbf {p} _ {p} _ {β_c} $ r \ geq 1 $，其中$λ_r= [ - r，r]^d \ cap \ mathbb {z}^d $。换句话说，$ \ mathbb {z}^d $上的关键两点函数始终通过层次晶格上的关键两点函数在上面的平均界定。据信，该上限对$α$的值严格低于跨界值$α_c（d）$是尖锐的，在$ \ mathbb {z}^d $和层次结构上，几种关键指数的值和层次结构相等。
Consider long-range Bernoulli percolation on $\mathbb{Z}^d$ in which we connect each pair of distinct points $x$ and $y$ by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $α>0$ is fixed and $β\geq 0$ is a parameter. We prove that if $0<α<d$ then the critical two-point function satisfies \[ \frac{1}{|Λ_r|}\sum_{x\in Λ_r} \mathbf{P}_{β_c}(0\leftrightarrow x) \preceq r^{-d+α} \] for every $r\geq 1$, where $Λ_r=[-r,r]^d \cap \mathbb{Z}^d$. In other words, the critical two-point function on $\mathbb{Z}^d$ is always bounded above on average by the critical two-point function on the hierarchical lattice. This upper bound is believed to be sharp for values of $α$ strictly below the crossover value $α_c(d)$, where the values of several critical exponents for long-range percolation on $\mathbb{Z}^d$ and the hierarchical lattice are believed to be equal.
