Long-range first-passage percolation on the complete graph

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 $\mathbb Z^d$.Comment: 16 page

Graph diameter in long-range percolation

We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show that the graph diameter for the graph reduced to a box of side $L$ scales like $(\log L)^{\Delta+o(1)}$ where $\Delta^{-1}:=\log_2(2d/s)$. In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance $L$. We also show that a ball of radius $r$ in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius $\exp\{r^{1/\Delta+o(1)}\}$ in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph diameter, substantially revised and corrected, added a result on volume growth asymptoti

Transience and recurrence of random walks on percolation clusters in an ultrametric space

We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)}, \delta>-1$, with $c_k=C_0+C_1\log k+C_2k^\alpha$, non-negative constants $C_0, C_1, C_2$, and $\alpha>0$. Percolation was proved in Dawson and Gorostiza (2013) for $\delta0$, with $\alpha>2$. In this paper we improve the result for the critical case by showing percolation for $\alpha>0$. We use a renormalization method of the type in the previous paper in a new way which is more intrinsic to the model. The proof involves ultrametric random graphs (described in the Introduction). The results for simple (nearest neighbour) random walks on the percolation clusters are: in the case $\delta<1$ the walk is transient, and in the critical case $\delta=1, C_2>0,\alpha>0$, there exists a critical $\alpha_c\in(0,\infty)$ such that the walk is recurrent for $\alpha<\alpha_c$ and transient for $\alpha>\alpha_c$. The proofs involve graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.Comment: 27 page

On the scaling of the chemical distance in long-range percolation models

We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{\Delta+o(1)},\qquad x,y\in C_{\infty}, |x-y|\to\infty, where \Delta^{-1} is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x-y|\to\infty. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of ``small-world'' phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.Comment: Published at http://dx.doi.org/10.1214/009117904000000577 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Structures in supercritical scale-free percolation

Scale-free percolation is a percolation model on $\mathbb{Z}^d$ which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience vs. recurrence for dimension 1 and 2 and give sufficient conditions for transience in dimension 3 and higher. Finally, we show the existence of a hierarchical structure for parameters where vertices have degrees with infinite variance and obtain bounds on the cluster density.Comment: Revised Definition 2.5 and an argument in Section 6, results are unchanged. Correction of minor typos. 29 pages, 7 figure

Long-range percolation on the hierarchical lattice

We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed $\beta$, we show that the critical value $\alpha_c(\beta)$ is non-trivial if and only if $N < \beta < N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $\alpha_c(\beta)$ as a function of $\beta$. This means that the phase diagram of this model is well understood.Comment: 24 page