Long-range first-passage percolation on the torus

We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph $\mathcal K_n$ are embedded in the $d$-dimensional torus $\mathbb T_n^d$, and each edge $e$ is assigned an independent transmission time $T_e=\|e\|_{\mathbb T_n^d}^\alpha E_e$, where $E_e$ is a rate-one exponential random variable associated with the edge $e$, $\|\cdot\|_{\mathbb T_n^d}$ denotes the torus-norm, and $\alpha\geq0$ is a parameter. We are interested in the case $\alpha\in[0,d)$, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $\mathbb Z^d$ studied by Chatterjee and Dey, and which extends first-passage percolation on the complete graph (the $\alpha=0$ case) studied by Janson. We consider the typical distance, flooding time, and diameter of the model. Our results show a $1,2,3$-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on $\mathbb Z^d$.