First Passage Percolation Model Research Articles

Limiting shape for first-passage percolation models on random geometric graphs

AbstractLet a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape, and we show that the shape is a Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. In the latter case we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

First-order behavior of the time constant in Bernoulli first-passage percolation

We consider the standard model of first-passage percolation on Zd (d≥2), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter 1−ε. These passage times induce a random pseudo-metric Tε on Rd. By subadditive arguments, it is well known that for any z∈Rd∖{0}, the sequence Tε(0,nz)/n converges a.s. toward a constant με(z) called the time constant. We investigate the behavior of ε↦με(z) near 0, and prove that με(z)=‖z‖1−C(z)ε1/d1(z)+o(ε1/d1(z)), where d1(z) is the number of nonnull coordinates of z, and C(z) is a constant whose dependence on z is partially explicit.

Continuity of the time constant in a continuous model of first passage percolation

Etant donnés une dimension d≥2 et une mesure finie ν sur (0,+∞), nous considérons ξ un processus ponctuel de Poisson sur Rd×(0,+∞) de mesure d’intensité dc⊗ν où dc désigne la mesure de Lebesgue sur Rd. Nous considérons le modèle booléen Σ=⋃ (c,r)∈ξB(c,r) où B(c,r) désigne la boule ouverte centrée en c et de rayon r. Pour tous x,y∈Rd, nous définissons T(x,y) comme le temps minimal nécessaire pour voyager de x à y pour un voyageur qui se déplace à vitesse 1 en dehors de Σ et à vitesse infinie dans Σ. Par une application standard du théorème ergodique sous-additif de Kingman, on peut facilement prouver que T(0,x) se comporte comme μ‖x‖ quand ‖x‖ tend vers l’infini, où μ est une constante appelée la constante de temps en percolation de premier passage classique. Dans cet article, nous étudions la régularité de μ comme fonction de la mesure ν associée au modèle booléen sous-jacent. Un des résultats clés est un contrôle uniforme de la longueur de “bonnes” géodésiques. Au cours de la preuve, nous avons recours à un analogue continu de l’inégalité BK pour des unions d’occurrences disjointes d’évènements.

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.

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.

Shape of shortest paths in random spatial networks.

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^{χ}, while the transversal fluctuations, known as wandering, grow as L^{ξ}. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ=3/5 and χ=1/5, or ξ=7/10 and χ=2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel-time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first-passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbor random geometric graphs, where via Monte Carlo simulations we find ξ=0.60±0.01 and χ=0.20±0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β skeletons and the Delaunay triangulation and are characterized by the values ξ=0.70±0.01 and χ=0.40±0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the so-called Kardar-Parisi-Zhang (KPZ) relation χ=2ξ-1 is satisfied for all these models. These results shed some light on the Euclidean first-passage process but also raise some theoretical questions about the scaling laws and the derivation of the exponent values and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

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.

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.

Kardar–Parisi–Zhang universality in first-passage percolation: the role of geodesic degeneracy

We have characterized the scaling behavior of the first-passage percolation (FPP) model on two types of discrete networks, the regular square lattice and the disordered Delaunay lattice, thereby addressing the effect of the underlying topology. Several distribution functions for the link-times were considered. The asymptotic behavior of the fluctuations for both the minimal arrival time and the lateral deviation of the geodesic path are in perfect agreement with the Kardar–Parisi–Zhang (KPZ) universality class regardless of the type of the link-time distribution and of the lattice topology. Pre-asymptotic behavior, on the other hand, is found to depend on the uniqueness of geodesics in absence of disorder in the local crossing times, a topological property of lattice directions that we term geodesic degeneracy. This property has important consequences on the model, as for example the well-known anisotropic growth in regular lattices. In this work we provide a framework to understand its effect as well as to characterize its extent.

Positivity of the time constant in a continuous model of first passage percolation

For a given dimension d $\ge$ 2 and a finite measure $\nu$ on (0, +$\infty$), we consider $\xi$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $\nu$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $\Sigma$ = $\cup$ (c,r)$\in$$\xi$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $\Sigma$ and at infinite speed inside $\Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $\mu$ x when x goes to infinity, where $\mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $\mu$ as a function of the measure $\nu$ associated with the underlying Boolean model.

Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes

Author(s): Aldous, DJ | Abstract: A simple lemma bounds s.d.(T )/ET for hitting times T in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result concerns a model of first passage percolation on a finite graph, where the traversal times of edges are independent Exponentials with arbitrary rates. Consider the percolation time X between two arbitrary vertices. We prove that s.d.(X)/EX is small if and only if Ξ/EX is small, where Ξ is the maximal edge-traversal time in the percolation path attaining X.

Entropy reduction in Euclidean first-passage percolation

The Euclidean first-passage percolation (FPP) model of Howard and Newman is a rotationally invariant model of FPP which is built on a graph whose vertices are the points of homogeneous Poisson point process. It was shown that one has (stretched) exponential concentration of the passage time $T_n$ from $0$ to $n\mathbf{e}_1$ about its mean on scale $\sqrt{n}$, and this was used to show the bound $\mu n \leq \mathbb{E}T_n \leq \mu n + C\sqrt{n} (\log n)^a$ for $a,C>0$ on the discrepancy between the expected passage time and its deterministic approximation $\mu = \lim_n \frac{\mathbb{E}T_n}{n}$. In this paper, we introduce an inductive entropy reduction technique that gives the stronger upper bound $\mathbb{E}T_n \leq \mu n + C_k\psi(n) \log^{(k)}n$, where $\psi(n)$ is a general scale of concentration and $\log^{(k)}$ is the $k$-th iterate of $\log$. This gives evidence that the inequality $\mathbb{E}T_n - \mu n \leq C\sqrt{\mathrm{Var}~T_n}$ may hold.

On some fundamental aspects of polyominoes on random Voronoi tilings

Consider a Voronoi tiling of $\mathbb{R} ^{d}$ based on a realization of an inhomogeneous Poisson random set. A Voronoi polyomino is a finite and connected union of Voronoi tiles. In this paper we provide tail bounds for the number of boxes that are intersected by a Voronoi polyomino, and vice-versa. These results will be crucial to analyze self-avoiding paths, greedy polyominoes and first-passage percolation models on Voronoi tilings and on the dual graph, named the Delaunay triangulation [Asymptotics for first-passage times on Delaunay triangulations (2011) Preprint, Greedy Polyominoes and first-passage times on random Voronoi tilings (2012) Preprint].

Greedy polyominoes and first-passage times on random Voronoi tilings

Let $\mathcal{N}$ be distributed as a Poisson random set on $\mathbb{R}^d$, $d\geq 2$, with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of $\mathbb{R}^d$, $\{C_v\}_{v\in \mathcal{N}}$, where $C_v$ is composed of points $\mathbf{x}\in\mathbb{R}^d$ that are closer to $v\in\mathcal{N}$ than to any other $v'\in\mathcal{N}$. A polyomino $\mathcal{P}$ of size $n$ is a connected union (in the usual $\mathbb{R}^d$ topological sense) of $n$ tiles, and we denote by $\Pi_n$ the collection of all polyominos $\mathcal{P}$ of size $n$ containing the origin. Assume that the weight of a Voronoi tile $C_v$ is given by $F(C_v)$, where $F$ is a nonnegative functional on Voronoi tiles. In this paper we investigate for some functionals $F$, mainly when $F(C_v)$ is a polynomial function of the number of faces of $C_v$, the tail behavior of the maximal weight among polyominoes in $\Pi_n$: $F_n=F_n(\mathcal{N}):=\max_{\mathcal{P}\in\Pi_n} \sum_{v\in \mathcal{P}} F(C_v)$. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance.

Asymptotics for First-Passage Times on Delaunay Triangulations

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

Continuous first-passage percolation and continuous greedy paths model : linear growth

We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting.

Continuous first-passage percolation and continuous greedy paths model: Linear growth

We study a random growth model on $\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting.

The time constant and critical probabilities in percolation models

We consider a first-passage percolation (FPP) model on a Delaunay triangulation $\mathcal{D}$ of the plane. In this model each edge $\mathbf{e}$ of $\mathcal{D}$ is independently equipped with a nonnegative random variable $\tau_\mathbf{e}$, with distribution function $\mathbb{F}$, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman [9] have shown that, under a suitable moment condition on $\mathbb{F}$, the minimum time taken to reach a point $\mathbf{x}$ from the origin $\mathbf{0}$ is asymptotically $\mu(\mathbb{F})|\mathbf{x}|$, where $\mu(\mathbb{F})$ is a nonnegative finite constant. However the exact value of the time constant $\mu(\mathbb{F})$ still a fundamental problem in percolation theory. Here we prove that if $\mathbb{F}(0)<1-p_c^*$ then $\mu(\mathbb{F})>0$, where $p_c^*$ is a critical probability for bond percolation on the dual graph $\mathcal{D}^*$.

Coexistence in two-type first-passage percolation models

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on ℤd or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x,y∈ℤd, there is a strictly positive probability that {z∈ℤd;d(y,z)<d(x,z)} and {z∈ℤd;d(y,z)>d(x,z)} are both infinite sets. We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by Häggström and Pemantle for independent exponential times on the square lattice.

Geodesics and Recurrence of Random Walks in Disordered Systems

In a first-passage percolation model on the square lattice $Z^2$, if the passage times are independent then the number of geodesics is either $0$ or $+\infty$. If the passage times are stationary, ergodic and have a finite moment of order $\alpha > 1/2$, then the number of geodesics is either $0$ or $+\infty$. We construct a model with stationary passage times such that $E\lbrack t(e)^\alpha\rbrack < \infty$, for every $0 < \alpha < 1/2$, and with a unique geodesic. The recurrence/transience properties of reversible random walks in a random environment with stationary conductances $( a(e);e$ is an edge of $\mathbb{Z}^2)$ are considered.