Limiting shape for directed percolation models

  title={Limiting shape for directed percolation models},
  author={James B. Martin},
  journal={Annals of Probability},
We consider directed first-passage and last-passage percolation on the nonnegative lattice Z d + , d ≥ 2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x) = lim n→ ∞ n- 1 T(|nx|) exist and are constant a.s. for x ∈ R d + , where T(z) is the passage time from the origin to the vertex z ∈ Z d + . We show that this shape function g is continuous on R d + , in particular at the boundaries. In two dimensions, we give more… 

Figures from this paper

On Heavy‐Tail Phenomena in Some Large‐Deviations Problems

  • F. Augeri
  • Mathematics
    Communications on Pure and Applied Mathematics
  • 2020
We revisit the proof of the large‐deviations principle of Wiener chaoses partially given by Borell and then by Ledoux in its full form. We show that some heavy‐tail phenomena observed in large

The scaling relation χ = 2 ξ − 1 for directed polymers in a random environment

We prove the scaling relation χ = 2ξ− 1 between the transversal exponent ξ and the fluctuation exponent χ for directed polymers in a random environment in d dimensions. The definition of these

Time constant for the once-oriented last passage site percolation in high dimensions

Let η be a real random variable whose logarithmic moment generating function λ(β) := ln(E exp(βη)) exists for all β > 0, and also such that E|η| < ∞. Let νd denote the point to line last passage time

Universality of Makespan in Flowshop Scheduling Problem

Makepan possesses universality in the sense of being little affected by a change in the probability distribution of the processing time, makespan can be decomposed into the sum of two shape functions, and makespan is less affected by the dispatching rule than by the scheduling procedure.

The scaling relation chi = 2 xi - 1 for directed polymers in a random environment

We prove the scaling relation chi = 2 xi - 1 between the transversal exponent xi and the fluctuation exponent chi for directed polymers in a random environment in d dimensions. The definition of

A directed polymer approach to once-oriented first passage site percolation in high dimensions

Let be a real valued random variable with a logarithmic moment generating function ( ): = lnEexp( ) for all 0. Consider a random environment {(z),z2 (d+1) }, for d 3, where the (z) are independent

Random growth models: Shape and convergence rate

  • M. Damron
  • Mathematics
    Proceedings of Symposia in Applied Mathematics
  • 2018
Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media.

Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights

These lecture notes discuss several related features of the exactly solvable two-dimensional corner growth model with exponentially distributed weights. A key property of this model is the



Linear growth for greedy lattice animals

Aspects of first passage percolation

Random matrices, non-colliding particle systems and queues

  • In Séminaire de Probabilités XXXVI, no. 1801 in Lecture Notes in Mathematics,
  • 2003

Inequalities for the Time Constant in First-Passage Percolation

Divergence of Shape Fluctuations in Two Dimensions

We consider stochastic growth models, such as standard first-passage percolation on Z d , where to leading order there is a linearly growing deterministic shape. Under natural hypotheses, we prove

On the non-convexity of the time constant in first-passage percolation.

We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions.

Greedy Lattice Animals II: Linear Growth

1. Outline. Since the description and the motivation of our model were already given in part I [Cox, Gandolfi, Griffin and Kesten (1993)], we just remind the reader of some definitions and jump right

Asymptotic results on infinite tandem queueing networks

Abstract. We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between