Related papers: First Passage percolation on a hyperbolic graph ad…

In 1993 van den Berg and Kesten proved a strict monotonicity theorem for first passage percolation on $\mathbb{Z}^d$, $d \ge 2$: given two probability measures $\nu$ and $\tilde{\nu}$ with finite mean, if $\tilde{\nu}$ is strictly more…

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly…

We consider first-passage percolation on the $d$ dimensional cubic lattice for $d \geq 2$; that is, we assign independently to each edge $e$ a nonnegative random weight $t_e$ with a common distribution and consider the induced random graph…

Let $X_1,..., X_n$ be independent, uniformly random points from $[0,1]^2$. We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points $n$…

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the…

For first passage percolation (FPP) on Euclidean lattices $\mathbb{Z}^d$ with $d\ge 2$, it is expected that the variance of the first passage time between two points grows sublinearly in the distance with a universal exponent strictly…

We study bond percolation for a family of infinite hyperbolic graphs. We relate percolation to the appearance of homology in finite versions of these graphs. As a consequence, we derive an upper bound on the critical probabilities of the…

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times…

The asymptotic study of percolation on finite transitive graphs is considered. Several questions and very few answers regarding percolation on finite graphs are presented.

We study first-passage percolation on $\mathbb Z^d$, $d\ge 2$, with independent weights whose common distribution is compactly supported in $(0,\infty)$ with a uniformly-positive density. Given $\epsilon>0$ and $v\in\mathbb Z^d$, which…

The Poisson clumping heuristic has lead Aldous to conjecture the value of the first passage percolation on the hypercube in the limit of large dimensions. Aldous' conjecture has been rigorously confirmed by Fill and Pemantle [Annals of…

We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple…

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. We show for $d \geq 2$ that if $\lambda$ is…

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann…

In this paper, we study the maximal edge-traversal time (simply we call maximal weight hereafter) on the optimal paths in the first passage percolation for several edge distributions, including the Pareto and Weibull distributions. It is…

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…

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if…

The theorem of Dekking and Host regarding tightness around the mean of first passage percolation on the binary tree, from the root to a boundary of a ball, is generalized to a class of graphs which includes all lattices in hyperbolic spaces…

We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves…

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [9]. We describe our results…