Annals of Probability

Random growth models with polygonal shapes

Abstract

We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.

Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 181-218.

Dates
First available in Project Euclid: 17 February 2006

https://projecteuclid.org/euclid.aop/1140191536

Digital Object Identifier
doi:10.1214/009117905000000512

Mathematical Reviews number (MathSciNet)
MR2206346

Zentralblatt MATH identifier
1090.60077

Citation

Gravner, Janko; Griffeath, David. Random growth models with polygonal shapes. Ann. Probab. 34 (2006), no. 1, 181--218. doi:10.1214/009117905000000512. https://projecteuclid.org/euclid.aop/1140191536

References

• Aigner, M. and Ziegler, G. M. (2001). Proofs from the Book, 2nd ed. Springer, New York.
• Alexander, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30–55.
• Bohman, T. and Gravner, J. (1999). Random threshold growth dynamics. Random Structures Algorithms 15 93–111.
• Bramson, M. and Gray, L. (1991). A useful renormalization argument. In Random Walks, Brownian Motion, and Interacting Particle Systems. Festshrift in Honor of Frank Spitzer (R. Durrett and H. Kesten, eds.) 113–152. Birkhäuser, Boston.
• Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Brooks-Cole, Belmont, MA.
• Durrett, R. and Liggett, T. M. (1981). The shape of the limit set in Richardson's growth model. Ann. Probab. 9 186–193.
• Eden, M. (1961). A two-dimensional growth process. Proc. Fourth Berkeley Symp. Math. Statist. Probab. 4 223–239. Univ. California Press, Berkeley.
• Gravner, J. (1999). Recurrent ring dynamics in two-dimensional excitable cellular automata. J. Appl. Probab. 36 492–511.
• Gravner, J. and Griffeath, D. (1993). Threshold growth dynamics. Trans. Amer. Math. Soc. 340 837–870.
• Gravner, J. and Griffeath, D. (1996). First passage times for discrete threshold growth dynamics. Ann. Probab. 24 1752–1778.
• Gravner, J. and Griffeath, D. (1997). Multitype threshold voter model and convergence to Poisson–Voronoi tessellation. Ann. Appl. Probab. 7 615–647.
• Gravner, J. and Griffeath, D. (1998). Cellular automaton growth on $\Bbb Z^2$: Theorems, examples and problems. Adv. in Appl. Math. 21 241–304.
• Gravner, J. and Griffeath, D. (1999). Reverse shapes in first-passage percolation and related growth models. In Perplexing Problems in Probability. Festshrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 121–142. Birkhäuser, Boston.
• Gravner, J., Tracy, C. and Widom, H. (2001). Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 1085–1132.
• Griffeath, D. (1981). The basic contact process. Stochastic Process. Appl. 11 151–186.
• Griffeath, D. Primordial soup kitchen. Available at http://psoup.math.wisc.edu.
• Hall, R. R. and Tenenbaum, G. (1988). Divisors. Cambridge Univ. Press.
• Hammersley, J. M. and Welsh, D. J. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Bernoulli, Bayes, Laplace Anniversary Volume (J. Neyman and L. LeCam, eds.) 61–110. Springer, New York.
• Janson, S., Luczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
• Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
• Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296–338.
• Kesten, H. and Schonmann, R. H. (1995). On some growth models with a small parameter. Probab. Theory Related Fields 101 435–468.
• Korshunov, A. D. and Shmulevich, I. (2002). On the distribution of the number of monotone Boolean functions relative to the number of lower units. Discrete Math. 257 463–479.
• Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71–95.
• Marchand, R. (2002). Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 1001–1038.
• Meakin, P. (1998). Fractals, Scaling and Growth far from Equilibrium. Cambridge Univ. Press.
• Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977–1005.
• Pimpinelli, A. and Villain, J. (1999). Physics of Crystal Growth. Cambridge Univ. Press.
• Richardson, D. (1973). Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 515–528.
• Seppäläinen, T. (1998). Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 26 1232–1250.
• Seppäläinen, T. (1999). Existence of hydrodynamics for the totally asymmetric simple $K$-exclusion process. Ann. Probab. 27 361–415.
• Sethna, J. Equilibrium crystal shapes. Available at http://www.lassp.cornell.edu/sethna/CrystalShapes/.
• Steele, J. M. (1997). Probability Theory and Combinatorial Optimization. SIAM, Philadelphia.
• Steele, J. M. and Zhang, Y. (2003). Nondifferentiability of the time constants of first-passage percolation. Ann. Probab. 31 1028–1051.
• Toom, A. L. (1974). Nonergodic multidimensional systems of automata. Probl. Inf. Transm. 10 239–246.
• Toom, A. L. (1980). Stable and attractive trajectories in multicomponent systems. In Advances in Probability and Related Topics (R. L. Dobrushin and Ya. G. Sinai, eds.) 6 549–575. Dekker, New York.
• Willson, S. J. (1978). On convergence of configurations. Discrete Math. 23 279–300.