Exact limiting shape for a simplified model of first-passage percolation on the plane
Timo Seppäläinen
Source: Ann. Probab. Volume 26, Number 3
(1998), 1232-1250.
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.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
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
References
[1] Alexander, K. S. (1993). A note on some rates of convergence in first-passage percolation. Ann. Appl. Probab. 3 81-90.
Mathematical Reviews (MathSciNet): MR94c:60167
Zentralblatt MATH: 0771.60090
Digital Object Identifier: doi:10.1214/aoap/1177005508
Project Euclid: euclid.aoap/1177005508
[2] Alexander, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30-55.
Mathematical Reviews (MathSciNet): MR98f:60203
Zentralblatt MATH: 0882.60090
Digital Object Identifier: doi:10.1214/aop/1024404277
Project Euclid: euclid.aop/1024404277
[3] Bardi, M. and Evans, L. C. (1984). On Hopf's formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal. 8 1373-1381.
Mathematical Reviews (MathSciNet): MR85k:35043
[4] Boivin, D. (1990). First-passage percolation: the stationary case. Probab. Theory Related Fields 86 491-499.
Mathematical Reviews (MathSciNet): MR92f:60177
Zentralblatt MATH: 0685.60103
Digital Object Identifier: doi:10.1007/BF01198171
[5] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583-603.
Zentralblatt MATH: 0462.60012
Mathematical Reviews (MathSciNet): MR624685
Digital Object Identifier: doi:10.1214/aop/1176994364
Project Euclid: euclid.aop/1176994364
[6] Durrett, R. and Liggett, T. (1981). The shape of the limit set in Richardson's growth model. Ann. Probab. 9 186-193.
Mathematical Reviews (MathSciNet): MR83f:60126
Zentralblatt MATH: 0457.60083
Digital Object Identifier: doi:10.1214/aop/1176994460
Project Euclid: euclid.aop/1176994460
[7] Evans, L. C. (1984). Some Min-Max methods for the Hamilton-Jacobi equation. Indiana Univ. Math. J. 33 31-50.
Mathematical Reviews (MathSciNet): MR726105
Zentralblatt MATH: 0543.35012
Digital Object Identifier: doi:10.1512/iumj.1984.33.33002
[8] Ferrari, P. A. (1996). Limit theorems for tagged particles. Markov Processes Related Fields 2 17-40.
Mathematical Reviews (MathSciNet): MR97j:60184
Zentralblatt MATH: 0879.60109
[9] Fontes, L. and Newman, C. M. (1993). First-passage percolation for random colorings of Zd. Ann. Appl. Probab. 3 746-762.
Mathematical Reviews (MathSciNet): MR94k:60156a
Digital Object Identifier: doi:10.1214/aoap/1177005361
Project Euclid: euclid.aoap/1177005361
[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.
Mathematical Reviews (MathSciNet): MR33:6731
Zentralblatt MATH: 0143.40402
[11] H¨aggstr ¨om, O. and Meester, R. (1995). Asymptotic shapes for stationary first-passage percolation. Ann. Probab. 23 1511-1522.
Mathematical Reviews (MathSciNet): MR96m:60237
Zentralblatt MATH: 0852.60104
Digital Object Identifier: doi:10.1214/aop/1176987792
Project Euclid: euclid.aop/1176987792
[12] Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Probab. Theory Related Fields 108 153-170.
Mathematical Reviews (MathSciNet): MR98g:60182
Zentralblatt MATH: 0883.60091
Digital Object Identifier: doi:10.1007/s004400050105
[13] Kesten, H. (1986). Aspects of first-passage percolation. Lecture Notes in Math. 1180 125- 264. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR876084
Zentralblatt MATH: 0602.60098
[14] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338.
Mathematical Reviews (MathSciNet): MR94m:60205
Zentralblatt MATH: 0783.60103
Digital Object Identifier: doi:10.1214/aoap/1177005426
Project Euclid: euclid.aoap/1177005426
[15] Kingman, J. F. C. (1968). The ergodic theory of subadditive stochastic processes. J. Royal Statist. Soc. Ser. B 30 499-510.
Mathematical Reviews (MathSciNet): MR40:8114
JSTOR: links.jstor.org
[16] Kipnis, C. (1986). Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397-408.
Mathematical Reviews (MathSciNet): MR88a:60173
Zentralblatt MATH: 0601.60098
Digital Object Identifier: doi:10.1214/aop/1176992523
Project Euclid: euclid.aop/1176992523
[17] Lions, P. L. (1982). Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London.
Mathematical Reviews (MathSciNet): MR667669
Zentralblatt MATH: 0497.35001
[18] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977-1005.
Zentralblatt MATH: 0835.60087
Mathematical Reviews (MathSciNet): MR1349159
Digital Object Identifier: doi:10.1214/aop/1176988171
Project Euclid: euclid.aop/1176988171
[19] Rezakhanlou, F. (1994). Evolution of tagged particles in non-reversible particle systems. Comm. Math. Phys. 165 1-32.
Mathematical Reviews (MathSciNet): MR96g:82037
Zentralblatt MATH: 0811.60094
Digital Object Identifier: doi:10.1007/BF02099734
Project Euclid: euclid.cmp/1104271031
[20] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
Mathematical Reviews (MathSciNet): MR43:445
[21] Saada, E. (1987). A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab. 15 375-381.
Mathematical Reviews (MathSciNet): MR88f:60179
Zentralblatt MATH: 0617.60096
Digital Object Identifier: doi:10.1214/aop/1176992275
Project Euclid: euclid.aop/1176992275
[22] Sepp¨al¨ainen, T. (1996). A microscopic model for the Burgers equation and longest increasing subsequences. Electronic J. Probab. 1 1-51.
Mathematical Reviews (MathSciNet): MR97d:60162
Zentralblatt MATH: 0891.60093
[23] Sepp¨al¨ainen, T. (1998). Hydrodynamic scaling, convex duality, and asymptotic shapes of growth models. Markov Processes Related Fields 4 1-26.
Mathematical Reviews (MathSciNet): MR99e:60221
Zentralblatt MATH: 0906.60082
[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.
Mathematical Reviews (MathSciNet): MR93e:60199
