We 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).

One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ>t)∼t-α with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

We prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n, as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the nearest neighbour graph for d, K ≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $\mathbb{Z}^d$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster.As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation.We also prove a flat edge result in the case of dimension 2. Various examples are also given.

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

In first-passage percolation (FPP) models, the passage time Tℓ from the origin to the point ℓeℓ satisfies f(ℓ) := ETℓ = μℓ + o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.

We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

We introduce and study a novel type of first-passage percolation problem on where the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probability p and (1 — p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e. pc > p > 1 – pc. Furthermore, we show that as p ↑ pc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

We present some general results related to first-passage percolation. These results involve families of random variables arranged in independent subfamilies. Applications are given to the study of random networks and to first-passage percolation.

Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk)→ μ(U) whenever Uk converges weakly to U.

An upper bound for the asymptotic velocity in various directions of first-passage percolation on the square lattice is derived. In particular this gives a lower bound for the so-called time constant. The result is generalized to other lattices. Numerical examples are included.

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT, then there exist constants 0 < a, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).

From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn(c) <∞, where Nn(c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.

Let μ (F) be the time constant of first-passage percolation on the square lattice with underlying distribution function F. Two theorems are presented which show, under some restrictions, that μ varies continuously in F with respect to weak convergence. These results are improvements of existing continuity theorems.

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

In this paper the global behaviour of percolation processes on the d-dimensional square lattice is studied. Using techniques of Richardson (1973) we prove, under weak moment assumptions on the time coordinate distribution, the following result. There exists a norm N(·) on Rd such that, for all 0 < ε < 1, we have that almost surely for all sufficiently large t the N-ball of radius (1 – ε)t is contained in η, (the set of all sites occupied by time t) and η, is contained in the N-ball of radius (1 + ε)t. Richardson (1973) derived the corresponding ‘in probability' result for a class of spread processes on Rd, satisfying certain conditions.

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.