Limiting shape for directed percolation models

By James B. Martin. 2003

89 citations recorded by the source corpus.

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 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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9e1123a1aab8b984f43b216d25c781966b84d509

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In Condensed Matter Physics

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Citations in the corpus

A GUE central limit theorem and universality of directed first andlast passage site percolation (2004, unknown): “Glynn and Whitt [7] proved that for k = o(N), (11) lim N,k→∞ L(N, k)− μN √ Nk = α, P-almost surely for some α, where α was proved to be equal to 2σ by Seppäläinen [17] (see also [11]).”
Stochastic Interface Models (2005, background): “See [197] for the directed first-passage and last-passage percolation models.”
50 years of first passage percolation (2015, background): “The details for the proof of the limit shape can be found in [133], including a necessary extra step to handle the continuity of Bν up to the boundary of the upper right quadrant.”
Stationary cocycles and Busemann functions for the corner growth model (2015, background): “ +2σ √ st. The Durrett-Liggett ﬂat edge result ([21], Theorem 3.3 below) tells us that this formula is not true for all i.i.d. weights. It does hold for general weights asymptotically at the boundary [43]: gpp(1,t)=m0 +2σ √ t+o( √ t) as t ց0. 2.3 Gradients and convexity Regularity properties of gpp ”
A Universality Property for Last-Passage Percolation Paths Close to the Axis (2004, unknown): “A diﬀerent sort of universality result for paths near the axis in directed percolation models is given in [13].”