The Annals of Probability

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

Timo Seppäläinen

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab. Volume 26, Number 3 (1998), 1232-1250.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1022855751

Mathematical Reviews number (MathSciNet)
MR1640344

Digital Object Identifier
doi:10.1214/aop/1022855751

Zentralblatt MATH identifier
0935.60093

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 82C22: Interacting particle systems [See also 60K35]

Keywords
First-passage percolation hydrodynamic limit tagged particle asymptotic shape

Citation

Seppäläinen, Timo. Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 26 (1998), no. 3, 1232--1250. doi:10.1214/aop/1022855751. http://projecteuclid.org/euclid.aop/1022855751.


Export citation

References

  • [1] Alexander, K. S. (1993). A note on some rates of convergence in first-passage percolation. Ann. Appl. Probab. 3 81-90.
  • [2] Alexander, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30-55.
  • [3] Bardi, M. and Evans, L. C. (1984). On Hopf's formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal. 8 1373-1381.
  • [4] Boivin, D. (1990). First-passage percolation: the stationary case. Probab. Theory Related Fields 86 491-499.
  • [5] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583-603.
  • [6] Durrett, R. and Liggett, T. (1981). The shape of the limit set in Richardson's growth model. Ann. Probab. 9 186-193.
  • [7] Evans, L. C. (1984). Some Min-Max methods for the Hamilton-Jacobi equation. Indiana Univ. Math. J. 33 31-50.
  • [8] Ferrari, P. A. (1996). Limit theorems for tagged particles. Markov Processes Related Fields 2 17-40.
  • [9] Fontes, L. and Newman, C. M. (1993). First-passage percolation for random colorings of Zd. Ann. Appl. Probab. 3 746-762.
  • [10] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Bernoulli-BayesLaplace Anniversary Volume (J. Neyman and L. Le Cam, eds.) 61-110. Springer, Berlin.
  • [11] H¨aggstr ¨om, O. and Meester, R. (1995). Asymptotic shapes for stationary first-passage percolation. Ann. Probab. 23 1511-1522.
  • [12] Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Probab. Theory Related Fields 108 153-170.
  • [13] Kesten, H. (1986). Aspects of first-passage percolation. Lecture Notes in Math. 1180 125- 264. Springer, Berlin.
  • [14] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338.
  • [15] Kingman, J. F. C. (1968). The ergodic theory of subadditive stochastic processes. J. Royal Statist. Soc. Ser. B 30 499-510.
  • [16] Kipnis, C. (1986). Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397-408.
  • [17] Lions, P. L. (1982). Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London.
  • [18] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977-1005.
  • [19] Rezakhanlou, F. (1994). Evolution of tagged particles in non-reversible particle systems. Comm. Math. Phys. 165 1-32.
  • [20] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • [21] Saada, E. (1987). A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab. 15 375-381.
  • [22] Sepp¨al¨ainen, T. (1996). A microscopic model for the Burgers equation and longest increasing subsequences. Electronic J. Probab. 1 1-51.
  • [23] Sepp¨al¨ainen, T. (1998). Hydrodynamic scaling, convex duality, and asymptotic shapes of growth models. Markov Processes Related Fields 4 1-26.
  • [24] Vahidi-Asl, M. Q. and Wierman, J. (1992). A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs 2 (A. Frieze and T. Luczak, eds.) 247-262. Wiley, New York.