First-passage Percolation Research Articles

Nonhomogeneous Euclidean first-passage percolation and distance learning

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We define such a sample distance and prove the convergence, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

Fluctuations of transverse increments in two-dimensional first passage percolation

We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d. nonnegative, continuous, and have a finite moment generating function in a neighborhood of 0. We derive consequences about transverse increments of passage times, assuming the model satisfies certain properties. Approximately, the assumed properties are the following: We assume that the standard deviation of the passage time on scale r is of some order σ(r), and σ(r),r>0 grows approximately as a power of r. Also, the tails of the passage time distributions for distance r satisfy an exponential bound on a scale σ(r) uniformly over r. In addition, the boundary of the limit shape in a neighborhood of some fixed direction θ has a uniform quadratic curvature. By transverse increment we mean the difference between passage times from the origin to a pair of points which are approximately at the direction θ and the direction between the pair of points is the direction of the tangent to the boundary of the limit shape at the direction θ. The main consequence derived is the following. If σ(r) varies as rχ for some χ>0, and ξ is such that χ=2ξ−1, then the fluctuation of the transverse increment of passage time between a pair of points situated at distance r from each other is of the order of rχ∕ξ.

Maximal edge-traversal time in First-passage percolation

In this paper, we study the maximal edge-traversal time on optimal paths in First-passage percolation on the lattice Zd for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth depending on the tail of the edge distribution.

First-passage percolation in random planar maps and Tutte’s bijection

We consider large random planar maps and study the first-passage percolation distance obtained by assigning independent identically distributed lengths to the edges. We consider the cases of quadrangulations and of general planar maps. In both cases, the first-passage percolation distance is shown to behave in large scales like a constant times the usual graph distance. We apply our method to the metric properties of the classical Tutte bijection between quadrangulations with n faces and general planar maps with n edges. We prove that the respective graph distances on the quadrangulation and on the associated general planar map are in large scales equivalent when n→∞.

Percolation and first-passage percolation on oriented graphs

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices Zd that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

Fluctuation lower bounds in planar random growth models

Nous montrons des bornes inferieures de $\sqrt{\log n}$ pour l’ordre des fluctuations de trois modeles planaires de croissance (percolation de premier passage, percolation de dernier passage et polymeres diriges) sans autre hypothese sur la loi des poids des sommets ou des aretes que les conditions minimales permettant d’eviter les cas pathologiques. De telles bornes etaient connues auparavant seulement pour certaines classes restreintes de lois. De surcroit, nous montrons que l’exposant des fluctuations autour de la forme limite pour la percolation de premier passage est au moins $1/8$, ce qui etend des resultats precedents a des lois plus generales.

First-passage percolation under extreme disorder: From bond percolation to Kardar-Parisi-Zhang universality.

We consider the statistical properties of arrival times and balls on first-passage percolation (FPP) two-dimensional square lattices with strong disorder in the link times. A previous work showed a crossover in the weak disorder regime, between Gaussian and Kardar-Parisi-Zhang (KPZ) universality, with the crossover length decreasing as the noise amplitude grows. On the other hand, this work presents a very different behavior in the strong-disorder regime. An alternative crossover length appears below which the model is described by bond-percolation universality class. This characteristic length scale grows with the noise amplitude and diverges at the infinite-disorder limit. We provide a thorough characterization of the bond-percolation phase, reproducing its associated critical exponents through a careful scaling analysis of the balls, which is carried out through a continuous mapping of the FPP passage time into the occupation probability of the bond-percolation problem. Moreover, the crossover length can be explained merely in terms of properties of the link-time distribution. The interplay between the characteristic length and the correlation length intrinsic to bond percolation determines the crossover between the initial percolation-like growth and the asymptotic KPZ scaling.

Tertiles and the time constant

AbstractWe consider planar first-passage percolation and show that the time constant can be bounded by multiples of the first and second tertiles of the weight distribution. As a consequence, we obtain a counter-example to a problem proposed by Alm and Deijfen (2015).

Lower bounds for fluctuations in first-passage percolation for general distributions

En percolation de premier passage (PPP), on attribue des poids i.i.d. aux aretes du reseau cubique $\mathbb{Z}^{d}$ et analyse la metrique de graphe induite. Si $T(x,y)$ denote la distance entre les sommets $x$ et $y$, une question fondamentale est de trouver l’ordre des fluctuations de $T(0,x)$. Il est escompte que la variance de $T(0,x)$ croit comme la norme, $\|x\|$, de $x$ a une puissance strictement inferieure a $1$, mais les meilleures bornes inferieures disponibles a ce jour (et seulement pour $d=2$) sont d’ordres logarithmiques. Ce resultat a ete demontre dans les annees 90 et il a connu peu d’amelioration depuis. Dans ce papier, nous abordons le probleme d’obtenir des bornes inferieures plus strictes, en montrant qu’avec tres grande probabilite, la distance $T(0,y)$ n’est pas contenue dans un interval de taille $o(\log \|x\|)^{1/2}$. Un resultat similaire est aussi valide en percolation de dernier passage (PDP) dans des cylindres minces. Ce type de resultats qui n’avait ete obtenu que pour des classes particulieres de poids est ici demontre en toute generalite. Les (nouvelles) methodes developpees ici consistent a induire une fluctuation du nombre d’aretes dans une boite dont les poids sont en “mode-haut.”

Sublinear variance in Euclidean first-passage percolation

The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and ne1 obeys a diffusive upper bound: VarT(0,ne1)≤Cn, and in this paper we improve this inequality to Cn∕logn. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik–Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the passage time as a function of Bernoulli sequences and uniform sequences, and develop several “greedy lattice animal” arguments.

A general method for lower bounds on fluctuations of random variables

There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general method for lower bounds on fluctuations. The method is used to obtain new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington–Kirkpatrick model of spin glasses, first-passage percolation and random matrices. A long list of open problems is provided at the end.

Divergence of shape fluctuation for general distributions in first-passage percolation

Nous étudions les fluctuations de la forme limite pour la percolation de premier passage dans $\mathbb{Z}^{d}$. Il est connu que ces fluctuations divergent dans le cas des lois de Bernoulli [Zhang (Probab. Theory. Related. Fields. 136 (2006) 298–320)]. Dans cet article, nous étendons ce résultat à toutes les lois.

On scaling limit of a cost in adhoc network model

We are interested in giving a mathematical formula of a cost in adhoc network model. In our model, the cost is formulated as an application of first-passage percolation and the motion of devices is random, and an asymptotic density of devices is formulated by hydrodynamic limit. Under some technical assumptions, we give asymptotics of a cost in adhoc network model. In order to formulate this model, we extend the results of first-passage percolation given by Howard Newman [3] to that in inhomogeneous environments.

Asymptotics for 2D critical and near-critical first-passage percolation

We study Bernoulli first-passage percolation (FPP) on the triangular lattice in which sites have 0 and 1 passage times with probability p and $$1-p$$ , respectively. Denote by $${\mathcal {C}}_{\infty }$$ the infinite cluster with 0-time sites when $$p>p_c$$ , where $$p_c=1/2$$ is the critical probability. Denote by $$T(0,{\mathcal {C}}_{\infty })$$ the passage time from the origin 0 to $${\mathcal {C}}_{\infty }$$ . First we obtain explicit limit theorem for $$T(0,{\mathcal {C}}_{\infty })$$ as $$p\searrow p_c$$ . The proof relies on the limit theorem in the critical case, the critical exponent for correlation length and Kesten’s scaling relations. Next, for the usual point-to-point passage time $$a_{0,n}$$ in the critical case, we construct subsequences of sites with different growth rate along the axis. The main tool involves the large deviation estimates on the nesting of CLE $$_6$$ loops derived by Miller et al. (Ann Probab 44:1013–1052, 2016). Finally, we apply the limit theorem for critical Bernoulli FPP to a random graph called cluster graph, obtaining explicit strong law of large numbers for graph distance.

Liouville first-passage percolation: Subsequential scaling limits at high temperature

Let $\{Y_{\mathfrak{B}}(x):x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov–Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

Tight Fluctuations of Weight-Distances in Random Graphs with Infinite-Variance Degrees

We prove results for first-passage percolation on the configuration model with degrees having asymptotic finite mean, infinite variance and i.i.d. edge-weights with strictly positive support of the form Y=a+X, where a is a positive constant and the excess edge-weight X is a non-negative random variable with zero as the infimum of its support. We prove that the weight of the optimal path between two uniformly chosen vertices has tight fluctuations around the asymptotic mean of the graph-distance if and only if the following condition holds: the random variable X is such that the age-dependent branching process describing first-passage percolation exploration in the same graph with edge-weights from distribution X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost-shortest paths in the graph-distance proliferate, in the sense that there even exist paths having tight total excess edge-weight for appropriate edge-weight distributions. Our proof makes use of a degree-dependent percolation model that we believe is interesting in its own right, as well as tightness results for distances in scale-free configuration models that we prove to hold under rather weak conditions on the degrees.

First-passage percolation and local modifications of distances in random triangulations

We study local modifications of the graph distance in large random triangulations. Our main results show that, in large scales, the modified distance behaves like a deterministic constant $\mathbf{c}~\in~(0,\infty)$ times the usual graph distance. This applies in particular to the first-passage percolation distance obtained by assigning independent random weights to the edges of the graph. We also consider the graph distance on the dual map, and the first-passage percolation on the dual map with exponential edge weights, which is closely related to the so-called Eden model. In the latter two cases, we are able to compute explicitly the constant $\mathbf{c}$ by using earlier results about asymptotics for the peeling process. In general however, the constant $\mathbf{c}$ is obtained from a subadditivity argument in the infinite half-plane model that describes the asymptotic shape of the triangulation near the boundary of a large ball. Our results apply in particular to the infinite random triangulation known as the UIPT, and show that balls of the UIPT for the modified distance are asymptotically close to balls for the graph distance.

A temporal perspective on the rate of convergence in first-passage percolation under a moment condition

We study the rate of convergence in the celebrated Shape Theorem in first-passage percolation, obtaining the precise asymptotic rate of decay for the probability of linear order deviations under a moment condition. Our results are presented from a temporal perspective and complement previous work by the same author, in which the rate of convergence was studied from the standard spatial perspective.

Stationary coalescing walks on the lattice

We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In $2d$, we classify the collective behavior of these walks under mild assumptions: they either coalesce almost surely or form bi-infinite trajectories. Bi-infinite trajectories form measure-preserving dynamical systems, have a common asymptotic direction in $2d$, and possess other nice properties. We use our theory to classify the behavior of compatible families of semi-infinite geodesics in stationary first- and last-passage percolation. We also partially answer a question raised by C. Hoffman about the limiting empirical measure of weights seen by geodesics. We construct several examples: our main example is a standard first-passage percolation model where geodesics coalesce almost surely, but have no asymptotic direction or average weight.

Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation

Let us consider Euclidean first-passage percolation on the Poisson-Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an argument of Burton-Keane type and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.