We consider directed first-passage and last-passage percolation on the nonnegative lattice Z_+^d, d\geq2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x)=lim_{n\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in Z_+^d. We show that this shape function g is continuous on R_+^d, in particular at the boundaries. In two dimensions, we give more precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.
A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions. Corresponding results are obtained for the ground state of the `directed polymers in a random environment' model.
We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.
We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1.
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 Tracy-Widom law.
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the super-critical phase, which implies that the growth rate depends continuously on the percolation parameter.
We consider directed first passage percolation on the integer lattice, with time constant $μ$ and passage time $a_{0n}$ from the origin to $(n,0,...,0)$. It is shown that under certain conditions on the passage time distribution, $Ea_{0n} - nμ= O(n^{1/2}(\log\log n)/\log n)$.
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 $n^{(2κ+2)/(2κ+3)}\sqrt{\ln n}$, where $1\leκ<\infty$ is the curvature power of the shape function at $(1,1)$. The methodology of proof is robust enough to also apply to directed Bernoulli first passage site percolation, and further to longest common subsequences in random words.
A law of the iterated logarithm is established for the last passage times of directed percolation on rectangles in the plane over exponential or geometric independent random variables, rescaled to converge to the Tracy-Widom distribution. The normalization is of loglog type, in contrast with the log normalization for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result for the liminf is also discussed.
The sequence of random probability measures $ν_n$ that gives a path of length $n$, $\unsur{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.
In this note, we study the directed first passage percolation introduced in [F. Comets, R. Fukushima, S. Nakajima and N. Yoshida: Journal of Statistical Physics, 161-(3), 577-597 (2015)]. It is proved that the shortest path from the origin to the $n$-th hyperplane makes a jump larger than a small power of $\log n$. Some numerical results are also provided, which indicates that the maximal jump size is much larger in a certain parameter region.
For first passage percolation (FPP) on integer lattice with i.i.d. passage time distributions, in order to show existence of semi-infinite geodesics along a fixed direction, one requires unproven assumptions on the limiting shape. We consider FPP on two-dimensional integer lattice with i.i.d. passage times distributed as Durrett-Liggett class of measures. For this model, we show that along any direction in a deterministic angular sector (known as percolation cone), starting from every lattice point there exists an infinite geodesic along that direction and such directed geodesics coalesce almo…
Following the recent investigations of Baik and Suidan in \cite{baik2005gcl} and Bodineau and Martin in \cite{bodineau2005upl}, we prove large deviation properties for a last-passage percolation model in $\mathbb{Z}^{2}_{+}$ whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in \cite{baik2005gcl} and \cite{bodineau2005upl}, on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.
We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e.\ for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.
We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous…
We consider the directed first passage percolation model on ${\bf Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,θ))$ the passage time from the origin to $(r, θ)$ by a northeast path for $(r, θ)\in {\bf R}^+\times [0,π/2]$. It is known that $\vec{T}({\bf 0}, (r, θ))/r$ converges to a time constant $\vecμ_F (θ)$. Let $\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by $\vec{p}_c$, as follows: (1)…
The directed last passage percolation (LPP) on the quarter-plane is a growing model. To come into the growing set, a cell needs that the cells on its bottom and on its left to be in the growing set, and then to wait a random time. We present here a generalisation of directed last passage percolation (GLPP). In GLPP, the waiting time of a cell depends on the difference of the coming times of its bottom and left cells. We explain in this article the physical meaning of this generalisation. In this first work on GLPP, we study them as a growing model on the cylinders rather than on the quarter-pl…
We study site and bond percolation on directed simple random graphs with a given degree distribution and derive the expressions for the critical value of the percolation probability above which the giant strongly connected component emerges and the fraction of vertices in this component.
In this survey article we consider the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. We show how stationary cocycles are constructed from queueing fixed points and how these cocycles characterize the limit shape, yield existence of Busemann functions in directions where the shape has some regularity, describe the direction of the competition interface, and answer questions on existence, uniqueness, and coalescence of directional semi-infinite geodesics…
We discuss variational formulas for the limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under shifts, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measu…
The model of Brownian Percolation has been introduced as an approximation of discrete last-passage percolation models close to the axis. It allowed to compute some explicit limits and prove fluctuation theorems for these, based on the relations between the Brownian percolation and random matrices. Here, we present two approaches that allow to treat discrete asymmetric models of directed polymers. In both cases, the behaviour is universal, meaning that the results do not depend on the precise law of the environment as long as it satisfies some natural moment assumptions. First, we establish an …
We investigate extended processes given by last-passage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to non-universal and trivial extended processes.
We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.
Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on $Z^d$ where $d \ge 2$. The first new result is a type of uniqueness theorem: for every pair $x$ and $y$ of vertices which lie in infinite open paths, there exists almost surely a third vertex $z$ which is joined to infinity and which is attainable from $x$ and $y$ along directed open paths. Secondly, it is proved that a random walk on an infinite directed clust…
Suppose each site independently and randomly chooses some sites around it, and it is weakly (strongly) connected with them (if there choose each other). What is the probability that the weak (strong) connected cluster is infinite? We investigate a percolation model for this problem, which is a generalization of site percolation. We give a relation between the probability of the number of chosen sites around a site and the size of clusters. We also see the expected number of infinite clusters, and the exponential tail decay of the radius and the size of a cluster.
Previous studies of connectivity in wireless networks have focused on undirected geometric graphs. More sophisticated models such as Signal-to-Interference-and-Noise-Ratio (SINR) model, however, usually leads to directed graphs. In this paper, we study percolation processes in wireless networks modelled by directed SINR graphs. We first investigate interference-free networks, where we define four types of phase transitions and show that they take place at the same time. By coupling the directed SINR graph with two other undirected SINR graphs, we further obtain analytical upper and lower bound…
In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^α\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-α)/2}$, $0<α<1/3$.
We consider a directed percolation process on an ${\cal M}$ x ${\cal N}$ rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio $a = {\cal M}/{\cal N}$ at a value $a_c =[1-y^2-x(1-y)^2]/2y^2$ where P(x,y) is di…
In this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.
We study the time constant $C(ν)$ of last passage percolation on the complete directed acyclic graph on the set of non-negative integers, where edges have i.i.d. weights with distribution $ν$ with support included in $\{-\infty\}\cup\mathbb{R}$. We show that $ν\mapsto C(ν)$ is strictly increasing in $ν$. We also prove that $C(ν)$ is continuous in $ν$ for a large set of measures $ν$. Furthermore, when $ν$ is purely atomic, we show that $C(ν)$ is analytic with respect to the weights of the atoms. In the special case of two positive atoms, it is an explicit rational function of these weights.
In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants which is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.
