Limiting shape for directed percolation models

Abstract

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z_+^d, d\geq2, 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\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in 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 precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.

Published at http://dx.doi.org/10.1214/009117904000000838 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Shape Fluctuations and Random Matrices

Cited by in corpus (31)

• First passage percolation and competition models
• Geometry of geodesics through Busemann measures in directed last-passage percolation
• Joint distribution of Busemann functions in the exactly solvable corner growth model
• A shape theorem and semi-infinite geodesics for the Hammersley model with random weights
• Busemann functions and Gibbs measures in directed polymer models on ${\mathbb{Z}}^2$
• Directed last passage percolation with discontinuous weights
• Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights
• Limit shapes for inhomogeneous corner growth models with exponential and geometric weights
• A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation
• Stationary cocycles for the corner growth model
• Properties of the limit shape for some last passage growth models in random environments
• The scaling relation chi = 2 xi - 1 for directed polymers in a random environment
• Stochastic domination for the last passage percolation model
• Asymmetric directed polymers in random environments
• Busemann functions and Equilibrium measures in last passage percolation models
• Hydrodynamical Methods in Last Passage Percolation Models
• Geodesics and the competition interface for the corner growth model
• Stationary cocycles and Busemann functions for the corner growth model
• On The Time Constant for Last Passage Percolation on Complete Graph
• Sublinear variance for directed last-passage percolation
• An Oriented Competition model on Z_{+}^2
• Limit theorems for a random directed slab graph
• Variational formulas and cocycle solutions for directed polymer and percolation models
• Dissertation: Geodesics of Random Riemannian Metrics
• Non-existence of bi-infinite polymer Gibbs measures
• Universality of Makespan in Flowshop Scheduling Problem
• Heavy tails in last-passage percolation
• Properties of the limit shape for some last passage growth models in random environments (Dissertation)
• A universality property for last-passage percolation paths close to the axis
• Last Passage Percolation in Macroscopically Inhomogeneous Media
• Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle