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Distances in 1/∥x - y∥2d Percolation Models for all Dimensions

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DOI: 10.1007/s00220-023-04861-z

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This paper studies independent long-range percolation on Z(d), revealing the growth patterns of graph distance and box diameter, as well as their interconnection.
We study independent long-range percolation on Z(d) for all dimensions d, where the vertices u and v are connected with probability 1 for parallel to u - v parallel to(infinity) = 1 and with probability p(beta, {u, v}) = 1- e (-beta integral u+[0,1)d integral v+[0,1)d 1/parallel to x- y parallel to 22d dxdy) approximate to beta/parallel to u - v parallel to(2d)(2) for parallel to u - v parallel to(infinity) >= 2. Let u is an element of Z(d) be a point with parallel to u parallel to(infinity) = n. We show that both the graph distance D(0, u) between the origin 0 and u and the diameter of the box {0,..., n}(d) growlike n(theta(beta)), where 0 < theta(beta) < 1. We also show that the graph distance and the diameter of boxes have the same asymptotic growth when two vertices u, v with parallel to u - v parallel to(2) > 1 are connected with a probability that is close enough to p(beta, {u, v}). Furthermore, we determine the asymptotic behavior of theta(beta) for large beta, and we discuss the tail behavior of D(0,u)/parallel to u parallel to(theta(beta))(2).

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