Distância química em modelos de percolação de longo alcance

Abstract

This text shows results in Long-Range Percolation on the d-dimensional lattice Zd, d ≥ 1, which all nearest-neighbor edges are presents and each pair of vertices x, y ∈ Z d , with ||x − y|| > 1, are connected, independently of the others, with probability px,y ∈ [0, 1]. We assume that px,y is translation invariant and has polynomial decay px,y = ||x−y||−s+o(1) , s > 0. These results relate the chemical distance (graph theoretical distance) D(x, y) on the random graph with the euclidean distance ||x − y||, according to the parameter s.