Related papers: A Central Limit Theorem for First Passage Percolat…

We present a novel computational method of first-passage times between a starting site and a target site of regular bounded lattices. We derive accurate expressions for all the moments of this first-passage time, validated by numerical…

We investigate a limit theorem on traversable length inside semi-cylinder in the 2-dimensional supercritical Bernoulli bond percolation, which gives an extension of Theorem 2 in Grimmett(1981). This type of limit theorems was originally…

We consider first passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen…

This paper studies the first passage percolation (FPP) model: each edge in the cubic lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices…

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…

In 2003, Kahn conjectured a characterization of the critical percolation probability $p_c$ in terms of vertex cut sets (\cite{JK03}). Later, Lyons and Peres (2016) conjectured a similar characterization of $p_c$ , but in terms of edge cut…

Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a percolation threshold if for every $\varepsilon > 0$, the proportion $\left\lVert K_1…

Schramm's Locality Conjecture asserts that the value of the critical percolation parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this note, we prove this conjecture in the particular case of transitive…

We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the…

We consider a simple random walk on the T-fractal and we calculate the exact mean time $\tau^g$ to first reach the central node $i_0$. The mean is performed over the set of possible walks from a given origin and over the set of starting…

Consider an independent site percolation model with parameter $p \in (0,1)$ on $\Z^d,\ d\geq 2$ where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to each coordinate axis. We show that the percolation…

Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$…

We study the distribution of finite clusters in slightly supercritical ($p \downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if $G$ is a transitive nonamenable graph satisfying the…

We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through…

In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^{\alpha}\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the…

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 investigate first passage percolation on inhomogeneous random graphs. The random graph model G(n,kappa) we study is the model introduced by Bollob\'as, Janson and Riordan, where each vertex has a type from a type space S and edge…

We study the independent alignment percolation model on $\mathbb{Z}^d$ introduced by Beaton, Grimmett and Holmes [arXiv:1908.07203]. It is a model for random intersecting line segments defined as follows. First the sites of $\mathbb{Z}^d$…

We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has…

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…