First-passage Percolation Research Articles

Geodesics cross any pattern in first-passage percolation without any moment assumption and with possibly infinite passage times

In first-passage percolation, one places nonnegative i.i.d. random variables (T(e)) on the edges of Zd. A geodesic is an optimal path for the passage times T(e). Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. When we assume that the common distribution of the passage times satisfies a suitable moment assumption, it is shown in [Antonin Jacquet. Geodesics in first-passage percolation cross any pattern, arXiv:2204.02021, 2023] that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic. This paper completes this study by showing that this result remains true when we consider distributions with an unbounded support without any moment assumption or distributions with possibly infinite passage times. The techniques of proof differ from the preceding article and rely on a notion of penalized geodesic.

Long-Range First-Passage Percolation on the Torus

We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {K}_n$$\\end{document} are embedded in the d-dimensional torus Tnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb T_n^d$$\\end{document}, and each edge e is assigned an independent transmission time Te=‖e‖TndαEe\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_e=\\Vert e\\Vert _{\\mathbb T_n^d}^\\alpha E_e$$\\end{document}, where Ee\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_e$$\\end{document} is a rate-one exponential random variable associated with the edge e, ‖·‖Tnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\cdot \\Vert _{\\mathbb T_n^d}$$\\end{document} denotes the torus-norm, and α≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\ge 0$$\\end{document} is a parameter. We are interested in the case α∈[0,d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in [0,d)$$\\end{document}, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document} studied by Chatterjee and Dey [14], and which extends first-passage percolation on the complete graph (the α=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =0$$\\end{document} case) studied by Janson [24]. We consider the typical distance, flooding time, and diameter of the model. Our results show a 1, 2, 3-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document}.

Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

Shape effects in the fluctuations of random isochrones on a square lattice.

We consider the isochrone curves in first-passage percolation on a 2D square lattice, i.e., the boundary of the set of points which can be reached in less than a given time from a certain origin. The occurrence of an instantaneous average shape is described in terms of its Fourier components, highlighting a crossover between a diamond and a circular geometry as the noise level is increased. Generally, these isochrones can be understood as fluctuating interfaces with an inhomogeneous local width which reveals the underlying lattice structure. We show that once these inhomogeneities have been taken into account, the fluctuations fall into the Kardar-Parisi-Zhang universality class with very good accuracy, where they reproduce the Family-Vicsek Ansatz with the expected exponents and the Tracy-Widom histogram for the local radial fluctuations.

Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation

We consider first-passage percolation on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{Z}^{2}$\\end{document} 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 more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.The result leads to a quantitative resolution of the Benjamini–Kalai–Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge.We further prove that the limit shape assumption is satisfied for a specific family of distributions.Lastly, related to the 1965 Hammersley–Welsh highways and byways problem, we prove that the expected fraction of the square {−n,…,n}2 which is covered by infinite geodesics starting at the origin is at most an inverse power of n. This result is obtained without explicit limit shape assumptions.

Measure-valued growth processes in continuous space and growth properties starting from an infinite interface

The k-parent and infinite-parent spatial Lambda-Fleming Viot processes (or SLFV), introduced in Louvet (2023), form a family of stochastic models for spatially expanding populations. These processes are akin to a continuous-space version of the classical Eden growth model (but with local backtracking of the occupied area allowed when k is finite), while being associated with a dual process encoding ancestry.In this article, we focus on the growth properties of the area occupied by individuals of type 1 (type 0 encoding units of empty space). To do so, we first define the quantities that we shall use to quantify the speed of growth of the occupied area. Using the associated dual process and a comparison with a first-passage percolation problem, we show that the growth of the occupied region in the infinite-parent SLFV is linear in time. Because of the possibility of local backtracking of the occupied area, the result we obtain for the k-parent SLFV is slightly weaker. It gives an upper bound on the probability that a given location is occupied at time t, which also shows that growth in the k-parent SLFV is linear in time. We use numerical simulations to approximate the growth speed for the infinite-parent SLFV, and we observe that the actual speed may be higher than the speed expected from simple first-moment calculations due to the characteristic front dynamics.

Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation

This monograph resolves—in a dense class of cases—several open problems concerning geodesics in i.i.d. first-passage percolation on Z d \mathbb {Z}^d . Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 0 to n ξ n{\boldsymbol \xi } , where ξ {\boldsymbol \xi } is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these empirical measures converge weakly to a deterministic limit as n → ∞ n\to \infty , answering a question of Hoffman. These families include arbitrarily small L ∞ L^\infty -perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms can have any prescribed sequence of probabilities. Moreover, the constructions are explicit enough to guarantee examples possessing certain features, for instance: both continuous and discrete distributions whose support is all of [ 0 , ∞ ) [0,\infty ) , and distributions given by a density function that is k k -times differentiable. All results also hold for ξ {\boldsymbol \xi } -directed infinite geodesics. In comparison, we show that if Z d \mathbb {Z}^d is replaced by the infinite d d -ary tree, then any distribution for the weights admits a unique limiting empirical measure along geodesics. In both the lattice and tree cases, our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach also allows us to obtain new convergence results for geodesic lengths, which have been unimproved in the subcritical regime since the seminal 1965 manuscript of Hammersley and Welsh.

Fluctuation bounds for first-passage percolation on the square, tube, and torus

Fluctuation bounds for first-passage percolation on the square, tube, and torus

On the number and size of holes in the growing ball of first-passage percolation

First-passage percolation is a random growth model defined on Z d \mathbb {Z}^d using i.i.d. nonnegative weights ( τ e ) (\tau _e) on the edges. Letting T ( x , y ) T(x,y) be the distance between vertices x x and y y induced by the weights, we study the random ball of radius t t centered at the origin, B ( t ) = { x ∈ Z d : T ( 0 , x ) ≤ t } \mathbf {B}(t) = \{x \in \mathbb {Z}^d : T(0,x) \leq t\} . It is known that for all such τ e \tau _e , the number of vertices (volume) of B ( t ) \mathbf {B}(t) is at least order t d t^d , and under mild conditions on τ e \tau _e , this volume grows like a deterministic constant times t d t^d . Defining a hole in B ( t ) \mathbf {B}(t) to be a bounded component of the complement B ( t ) c \mathbf {B}(t)^c , we prove that if τ e \tau _e is not deterministic, then a.s., for all large t t , B ( t ) \mathbf {B}(t) has at least c t d − 1 ct^{d-1} many holes, and the maximal volume of any hole is at least c log ⁡ t c\log t . Conditionally on the (unproved) uniform curvature assumption, we prove that a.s., for all large t t , the number of holes is at most ( log ⁡ t ) C t d − 1 (\log t)^C t^{d-1} , and for d = 2 d=2 , no hole in B ( t ) \mathbf {B}(t) has volume larger than ( log ⁡ t ) C (\log t)^C . Without curvature, we show that no hole has volume larger than C t log ⁡ t Ct \log t .

A variational formula for large deviations in first-passage percolation under tail estimates

Consider the first passage percolation on the d-dimensional lattice Zd with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations P(T(0,nx)>n(μ+ξ)), for ξ>0 and x≠0 with a time constant μ, for weights that satisfy a tail assumption P(τe>t)≍βexp(−αtr). When r≤1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as exp(−(2dαξr+o(1))n). When 1<r≤d, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. The case r=d is critical and logarithmic corrections appear. For r∈(1,d), we show that the large deviation event {T(0,nx)>n(μ+ξ)} is described by a localization of high weights around the endpoints. The picture changes for r≥d where the configuration is not anymore localized.

Universality of the time constant for 2D critical first-passage percolation

Universality of the time constant for 2D critical first-passage percolation

Geodesic length and shifted weights in first-passage percolation

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 the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the ℓ 1 \ell ^1 distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produces singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.

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.

Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini and Tessera (Electron J Probab 20:1–20, 2015) and Cantrell and Furman (Groups Geom Dyn 11(4):1307–1345, 2017) show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any “conjugation-invariant” metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the “conjugation-invariant” condition is also a necessary condition in all cases where scaling limits are known to exist.

Absence of backward infinite paths for first-passage percolation in arbitrary dimension

In first-passage percolation (FPP), one places nonnegative random variables (weights) (te) on the edges of a graph and studies the induced weighted graph metric. We consider FPP on Zd for d≥2 and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of Furstenberg and most strongly supported for d=2 that, for continuously distributed i.i.d. weights, there a.s. are no bigeodesics. We provide the first progress on this question in general dimensions under no unproven assumptions. Our main result is that geodesic graphs, introduced in a previous paper of two of the authors, constructed in any deterministic direction a.s. do not contain doubly-infinite paths. As a consequence, one can construct random graphs of subsequential limits of point-to-hyperplane geodesics, which contain no bigeodesics. This gives evidence that bigeodesics, if they exist, cannot be constructed in a translation-invariant manner as limits of point-to-hyperplane geodesics.

Transitions for exceptional times in dynamical first-passage percolation

In first-passage percolation (FPP), we let $$(\tau _v)$$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of $$\tau _v$$ , there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if $$\sum F^{-1}(1/2+1/2^k) = \infty $$ , then a.s. there are exceptional times at which the weight grows atypically, but if $$\sum k^{7/8} F^{-1}(1/2+1/2^k) <\infty $$ , then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP.

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.

Comparison of limit shapes for Bernoulli first-passage percolation

We consider Bernoulli first-passage percolation on the [Formula: see text]-dimensional hypercubic lattice with [Formula: see text]. The passage time of edge [Formula: see text] is 0 with probability [Formula: see text] and 1 with probability [Formula: see text], independently of each other. Let [Formula: see text] be the critical probability for percolation of edges with passage time 0. When [Formula: see text], there exists a nonrandom, nonempty compact convex set [Formula: see text] such that the set of vertices to which the first-passage time from the origin is within [Formula: see text] is well approximated by [Formula: see text] for all large [Formula: see text], with probability one. The aim of this paper is to prove that for [Formula: see text], the Hausdorff distance between [Formula: see text] and [Formula: see text] grows linearly in [Formula: see text]. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the critical case.

First passage percolation on hyperbolic groups

We study first passage percolation (FPP) on a Gromov-hyperbolic group G with boundary ∂G equipped with the Patterson-Sullivan measure ν. We associate an i.i.d. collection of random passage times to each edge of a Cayley graph of G, and investigate classical questions about asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the ‘velocity’ exists in ν-almost every direction ξ∈∂G, and is almost surely constant by ergodicity of the G-action on ∂G. For every ξ∈∂G, we also show almost sure coalescence of any two geodesic rays directed towards ξ. Finally, we show that the variance of the first passage time grows linearly with word distance along word geodesic rays in every fixed boundary direction. This provides an affirmative answer to a conjecture in [15,14].

A Near-Optimal Rate of Periodic Homogenization for Convex Hamilton–Jacobi Equations

We consider a Hamilton-Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.