# Limiting shape for directed percolation models

@article{Martin2004LimitingSF, title={Limiting shape for directed percolation models}, author={James B. Martin}, journal={Annals of Probability}, year={2004}, volume={32}, pages={2908-2937} }

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…

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## References

SHOWING 1-10 OF 31 REFERENCES

### Random matrices, non-colliding particle systems and queues

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

### Divergence of Shape Fluctuations in Two Dimensions

- Mathematics
- 1995

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…

### Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions

- Mathematics
- 1981

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

- Mathematics
- 1996

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

- Mathematics
- 1994

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

- Mathematics
- 2000

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…