Johannes Baeumler: Chemical distances for long-range percolation


Consider long-range percolation on \mathbb{Z}^d, where there is an edge between two points x and y with probability asymptotic to \beta \|x-y\|^{-s}, independent of all other edges, for some positive parameters s and \beta. In this talk, we will focus on the metric properties of the long-range percolation graph. The chemical distance between two points x and y is the number of steps one needs to make in order to go from x to y. For different values of s, there are different regimes of how the chemical distance scales with the Euclidean distance. The transitions between these regimes happen at s=d and s=2d. After an overview of previous work, we will focus on the case s=2d. We will show that for s=2d, for each dimension d and for each \beta > 0, there exists a \theta=\theta(d,\beta) \in (0,1) such that the chemical distance between x and y is of order \|x-y\|^{\theta}. We will also discuss how the exponent \theta depends on the parameter \beta.


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