Tom Hutchcroft: Comparing critical long-range percolation on the Euclidean and hierarchical lattices

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 $\beta\geq 0$ is a parameter. We prove that 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 with the same d and alpha, whose asymptotics we computed in a previous paper. 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.

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Tom Hutchcroft: Comparing critical long-range percolation on the Euclidean and hierarchical lattices

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