Abstract:
Consider long-range Bernoulli percolation on in which we connect each pair of distinct points and by an edge with probability , where is fixed and is a parameter. We prove that the critical two-point function on 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 strictly below the crossover value , where the values of several critical exponents for long-range percolation on and the hierarchical lattice are believed to be equal.
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Based on