Related papers: First-passage percolation in random planar maps an…
We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight W_n and the number of edges H_n of the minimal weight path between two distinct…
We determine the asymptotic speed of the first-passage percolation process on some ladder-like graphs (or width-2 stretches) when the times associated with different edges are independent and exponentially distributed but not necessarily…
In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance $L$, geodesics exhibit deviations from their mean length $L$ that are of order $L^\chi$, while the transversal fluctuations, known as…
There are various models of first passage percolation (FPP) in $\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\mathbb Z^d$ to $\mathbb R^d$…
We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a…
We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\P[\tau=0]0$ is explicit) for the…
We study a large class of Bernoulli percolation models on random lattices of the half- plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold…
We consider the first passage percolation model on $\mathbf{Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. Let $T(u,v)$ be the passage time from $u$ to $v$. In this…
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…
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…
We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a…
In this paper we explore first passage percolation (FPP) on the Erd\H{o}s-R\'enyi random graph $G_n(p_n)$, where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when $np_n\to \lambda>1,$ we…
We consider the standard first passage percolation model on $\mathbb Z^d$ with bounded and bounded away from zero weights. We show that the rescaled passage time $\widetilde{\mathbf T}_{n,X}$ restricted to a compact set $X$ satisfies a…
We study non-random fluctuation in the first passage percolation on $\mathbb{Z}^d$ and show that it diverges for any dimension. We also prove the divergence of the non-random shape fluctuation, which was conjectured in [Yu Zhang. The…
We study the shape fluctuation in the first passage percolation on $\mathbb{Z}^d$. It is known that it diverges when the distribution obeys Bernoulli in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation.…
We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight…
Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a…
We construct the uniform infinite planar map (UIPM), obtained as the n \to \infty local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way…
Consider first passage percolation on $\mathbb{Z}^d$ with passage times given by i.i.d. random variables with common distribution $F$. Let $t_\pi(u,v)$ be the time from $u$ to $v$ for a path $\pi$ and $t(u,v)$ the minimal time among all…
The study of random graphs has become very popular for real-life network modeling such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice $\mathbb Z^d$, $d\ge1$, is a…