Large-range constant threshold growth model in one dimension

Gregor Sega (Faculty of Mathematics and Physics, University of Ljubljana)

Abstract


We study a one dimensional constant threshold model in continuous time. Its dynamics have two parameters, the range $n$ and the threshold $v$. An unoccupied site $x$ becomes occupied at rate 1 as soon as there are at least $v$ occupied sites in $[x-n, x+n]$. As n goes to infinity and $v$ is kept fixed, the dynamics can be approximated by a continuous space version, which has an explicit invariant measure at the front. This allows us to prove that the speed of propagation is asymptoticaly $n^2/2v$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 119-138

Publication Date: January 27, 2009

DOI: 10.1214/EJP.v14-598

References

  1. T. Bohman, J. Gravner. Random threshold growth dynamics. Random Structures Algorithms 15 (1999), 93-111. Math. Review 2000g:60155
  2. J.T. Cox and R. Durrett. Oriented percolation in dimensions $d \geq 4$: bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc. 93 (1983), 151-162. Math. Review 84e:60150
  3. H.A. David. Order Statistics, 2nd edition. Wiley, New York 1981. Math. Review 82i:62073
  4. R. Durrett and T.M. Liggett. The shape of the limit set in Richardson's growth model. Ann. Probab. 9 (1981), 186-193. Math. Review 83f:60126
  5. R. Fisch, J. Gravner and D. Griffeath. Metastability in the Greenberg-Hastings model. Ann. Appl. Probab. 3 (1993), 935-967. Math. Review 95f:60120
  6. J. Gravner and D. Griffeath. Random growth models with polygonal shapes. Ann. Probab. 34 (2006), 181-218. Math. Review 2007b:60237
  7. J. Gravner, C.A. Tracy and H. Widom. Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 26 (1998), 1085-1132. Math. Review 2002d:82065
  8. K. Johansson. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. 153 (2001), 259-296. Math. Review 2002g:05188
  9. S.P. Meyn and R.L. Tweedie. Markov chains and stochastic stability. Springer-Verlag, London 1993. Math. Review 95j:60103
  10. M.D. Penrose. Spatial epidemics with large finite range. J. Appl. Probab. 33 (1996), 933-939. Math. Review 97g:60134
  11. M.D. Penrose. The threshold contact process: a continuum limit. Probab. Theory Related Fields 104 (1996), 77-95. Math. Review 96j:60168
  12. T. Sepp‰l‰inen. Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 102 (2001), 1232-1250. Math. Review 99e:60220
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.