The Annals of Applied Probability

A phase transition for competition interfaces

Pablo A. Ferrari, James B. Martin, and Leandro P. R. Pimentel

Full-text: Open access

Abstract

We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle θ of the cone: for θ≥180°, the direction is deterministic, while for θ<180°, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 1 (2009), 281-317.

Dates
First available in Project Euclid: 20 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1235140340

Digital Object Identifier
doi:10.1214/08-AAP542

Mathematical Reviews number (MathSciNet)
MR2498679

Zentralblatt MATH identifier
1185.60109

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

Keywords
Asymmetric simple exclusion second-class particle Burgers equation rarefaction fan last-passage percolation competition interface

Citation

Ferrari, Pablo A.; Martin, James B.; Pimentel, Leandro P. R. A phase transition for competition interfaces. Ann. Appl. Probab. 19 (2009), no. 1, 281--317. doi:10.1214/08-AAP542. https://projecteuclid.org/euclid.aoap/1235140340


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References

  • [1] Balázs, M., Cator, E. and Seppäläinen, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 1094–1132 (electronic).
  • [2] Coletti, C. F. and Pimentel, L. P. R. (2007). On the collision between two PNG droplets. J. Stat. Phys. 126 1145–1164.
  • [3] Deijfen, M. and Häggström, O. (2006). The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model. Combin. Probab. Comput. 15 345–353.
  • [4] Derrida, B. and Dickman, R. (1991). On the interface between two growing Eden clusters. J. Phys. A 24, L191–L195.
  • [5] Ferrari, P. A. (1992). Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91 81–101.
  • [6] Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 820–832.
  • [7] Ferrari, P. A. and Fontes, L. R. G. (1994). Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Related Fields 99 305–319.
  • [8] Ferrari, P. A. and Kipnis, C. (1995). Second class particles in the rarefaction fan. Ann. Inst. H. Poincaré Probab. Statist. 31 143–154.
  • [9] Ferrari, P. A., Kipnis, C. and Saada, E. (1991). Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19 226–244.
  • [10] Ferrari, P. A., Martin, J. B. and Pimentel, L. P. R. (2006). Roughening and inclination of competition interfaces. Phys. Rev. E 73 031602.
  • [11] Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Probab. 33 1235–1254.
  • [12] Garet, O. and Marchand, R. (2005). Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15 298–330.
  • [13] Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 683–692.
  • [14] Heveling, M. and Last, G. (2005). Characterization of Palm measures via bijective point-shifts. Ann. Probab. 33 1698–1715.
  • [15] Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 739–747.
  • [16] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445–456.
  • [17] Kordzakhia, G. and Lalley, S. P. (2008). An oriented competition model on ℤ+2. Electron. Comm. Probab. 13 548–561.
  • [18] Kordzakhia, G. and Lalley, S. P. (2005). A two-species competition model on ℤd. Stochastic Process. Appl. 115 781–796.
  • [19] Godrèche, C., ed. (1992). Solids Far from Equilibrium. Collection Aléa-Saclay: Monographs and Texts in Statistical Physics 1. Cambridge Univ. Press, Cambridge.
  • [20] Martin, J. B. (2007). Geodesics in last-passage percolation and couplings of exclusion processes. In preparation.
  • [21] Mountford, T. and Guiol, H. (2005). The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15 1227–1259.
  • [22] Newman, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1017–1023. Birkhäuser, Basel.
  • [23] Pimentel, L. P. R. (2007). Multitype shape theorems for first passage percolation models. Adv. in Appl. Probab. 39 53–76.
  • [24] Rezakhanlou, F. (1995). Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 119–153.
  • [25] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41–53.
  • [26] Seppäläinen, T. (1998). Coupling the totally asymmetric simple exclusion process with a moving interface. Markov Process. Related Fields 4 593–628. I Brazilian School in Probability (Rio de Janeiro, 1997).
  • [27] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
  • [28] Wüthrich, M. V. (2002). Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane. In In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability 51 205–226. Birkhäuser Boston, Boston, MA.