LA Probability Forum

UCLA - Math Sciences Room 6627

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 $\beta$.