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- W4308205088 abstract "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 $1-exp(-beta|x-y|^{-d-alpha})$, where $alpha>0$ is fixed and $betageq 0$ is a parameter. We prove that if $0<alpha<d$ then the critical two-point function satisfies [ frac{1}{|Lambda_r|}sum_{xin Lambda_r} mathbf{P}_{beta_c}(0leftrightarrow x) preceq r^{-d+alpha} ] for every $rgeq 1$, where $Lambda_r=[-r,r]^d cap mathbb{Z}^d$. 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 $alpha$ strictly below the crossover value $alpha_c(d)$, where the values of several critical exponents for long-range percolation on $mathbb{Z}^d$ and the hierarchical lattice are believed to be equal." @default.
- W4308205088 created "2022-11-09" @default.
- W4308205088 creator A5060156386 @default.
- W4308205088 date "2022-11-01" @default.
- W4308205088 modified "2023-10-17" @default.
- W4308205088 title "Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on Zd" @default.
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- W4308205088 doi "https://doi.org/10.1063/5.0088450" @default.
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