**Abstract:**

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

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Based on

https://arxiv.org/abs/2208.04800

https://arxiv.org/abs/2208.04793