Johannes Baeumler: Chemical distances for long-range percolation

Abstract:

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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Based on

https://arxiv.org/abs/2208.04800

https://arxiv.org/abs/2208.04793

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Johannes Baeumler: Chemical distances for long-range percolation

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