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


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.


Based on


Password Protected

This video is password-protected. Please verify with a password to unlock the content.