First Passage Percolation Model Research Articles

Differentiability and monotonicity of expected passage time in Euclidean first-passage percolation

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.

Differentiability and monotonicity of expected passage time in Euclidean first-passage percolation

In first-passage percolation (FPP) models, the passage timeTℓfrom the origin to the pointℓeℓsatisfiesf(ℓ) := ETℓ= μℓ+o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously thatf(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 forf′(ℓ). In all dimensions, for certain values of the model's only parameter we havef′(ℓ) ≥C> 0 for largeℓ.

Lower bounds for point-to-point wandering exponents in Euclidean first-passage percolation

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

Lower bounds for point-to-point wandering exponents in Euclidean first-passage percolation

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

Geodesics and crossing Brownian motion in a soft Poissonian potential

We compare the model of crossing Brownian motion in a soft Poissonian potential with the model of continuum first-passage percolation among soft Poissonian obstacles. In both models we construct (via a shape theorem) a deterministic norm on R d , called the Lyapounov coefficient in the former and time-constant in the latter. The main theorem of this article claims that the properly rescaled Lyapounov coefficient converges to the time-constant as the strength of the potential tends to infinity.

Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment

Assign to the lattice sizes $z \epsilon \mathbb{Z}^d$ i.i.d. random 2 $d$-dimensional vectors $(\omega(z, z + e))_{|e|=1}$ whose entries take values in the open unit interval and add up to one. Given a realization $\omega$ of this environment, let $(X_n)_{n \geq o}$ be a Markov chain on $\mathbb{Z}^d$ which, when at $z$, moves one step to its neighbor $z + e$ with transition probability $\omega(z, z + e)$. We derive a large deviation principle for $X_n/n$ by means of a result similar to the shape theorem of first-passage percolation and related models. This result produces certain constants that are the analogue of the Lyapounov exponents known from Brownian motion in Poissonian potential or random walk in random potential. We follow a strategy similar to Sznitman.

Exact limiting shape for a simplified model of first-passage percolation on the plane

We derive the limiting shape for the following model of first-passage bond percolation on the two-dimensional integer lattice: the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions. The passage times of horizontal bonds are Bernoulli distributed, while the passage times of vertical bonds are all equal to a deterministic constant. To analyze the percolation model, we couple it with a one-dimensional interacting particle system. This particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far-away particles. We prove a law of large numbers for a tagged particle in this process, and the shape result for the percolation is obtained as a corollary.

Euclidean models of first-passage percolation

We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝ d . Compared to standard FPP on ℤ d , these models have some technical complications but also have the advantage of statistical isotropy. We prove two almost sure results: a shape theorem (where isotropy implies an exact Euclidean ball for the asymptotic shape) and nonexistence of certain doubly infinite geodesics (where isotropy yields a stronger result than in standard FPP).

On the number of infinite geodesics and ground states in disordered systems

We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to +∞. As corollaries we show that in any dimensiond≥2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or +∞. Proofs employ simple large-deviation estimates and ergodic arguments.

Contact propagation: Percolation and other scaling regimes.

We examine the scaling behavior of stirred percolation and first-passage percolation systems in which cluster contact is rate limiting with respect to propagation or transport. For the scaling regime near the static percolation threshold, we provide details of previously reported computational estimates of the ``propagation exponent,'' and we present results for several variants of the stirred percolation model and for a first-passage percolation model. Five additional scaling regimes are predicted, three of which are verified computationally. Applications to combustion, diffusion-limited aggregation, gasification of porous solids, and crack propagation in solids are discussed.

Weak moment conditions for time coordinates in first-passage percolation models

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.

Weak moment conditions for time coordinates in first-passage percolation models

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.