Related papers: Small deviation estimates and small ball probabili…

First-passage percolation is the study of the metric space $(\mathbb{Z}^d,T)$, where $T$ is a random metric defined as the weighted graph metric using random edge-weights $(t_e)_{e\in \mathcal{E}^d}$ assigned to the nearest-neighbor edges…

We study directed last-passage percolation on the planar square lattice whose weights have general distributions, or equivalently, queues in series with general service distributions. Each row of the last passage model has its own randomly…

In this note we give a(nother) combinatorial proof of an old result of Baik--Rains: that for appropriately considered independent geometric weights, the generating series for last passage percolation polymers in a $2n \times n \times n$…

We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on $\mathbb R^d$. We are motivated in our study by the random geometry of…

We study planar first-passage percolation with independent weights whose common distribution is supported in $(0,\infty)$ and is absolutely continuous with respect to Lebesgue measure. We prove that the passage time from $x$ to $y$ denoted…

Last three years have seen new developments in the theory of last passage percolation, which has variety applications to random permutations, random growth and random vicious walks. It turns out that a few class of models have determinant…

In this article, we prove an extreme value theorem on the limit distribution of geodesics in a geometrically finite quotient of $\Gamma\backslash\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits…

For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance $n$, is macroscopically…

Geodesic coalescence, or the tendency of geodesics to merge together, is a hallmark phenomenon observed in a variety of planar random geometries involving a random distortion of the Euclidean metric. As a result of this, the union of…

For directed Bernoulli last passage percolation with i.i.d.~weights on vertices over a $n\times n$ grid and for $n$ large enough, the geodesics are shown to be concentrated in a cylinder, centered on the main diagonal and of width of order…

We prove a GUE central limit theorem for random variables with finite fourth moment. We apply this theorem to prove that the directed first and last passage percolation problems in thin rectangles exhibit universal fluctuations given by the…

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about…

We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time t will have a width of order t^{1/3}. We…

Given a `cost' functional $F$ on paths $\gamma$ in a domain $D\subset\mathbb{R}^d$, in the form $F(\gamma) = \int_0^1 f(\gamma(t),\dot\gamma(t))dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_1,\ldots, X_n$…

We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $\Omega$ in $\mathbb R^d$. We denote by $\Gamma^1$ and $\Gamma^2$ two disjoint subsets of $\partial \Omega$…

In line with Pomeau's conjecture about the relevance of directed percolation (DP) to turbulence onset/decay in wall-bounded flows, we propose a minimal stochastic model dedicated to the interpretation of the spatially intermittent regimes…

We prove that the rescaled one-point fluctuations of the boundary of the percolation cluster in the Bernoulli-Exponential first passage percolation around the diagonal converge to a new family of distributions. The limit law is indexed by…

We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index alpha<2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape…

The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge $(i,j), |i-j|\neq 1 \mod n$ with probability $\rho/n$ for some $\rho>0$ constant. In this paper we add i.i.d. exponential edge weights…

We consider directed random graphs, the prototype of which being the Barak-Erd\H{o}s graph $\vec G(\mathbb Z, p)$, and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the graphs to certain…