Coexistence probability in the last passage percolation model is 6 - 8 log 2

David Coupier; Philippe Heinrich

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 973-988
  • ISSN: 0246-0203

Abstract

top
A competition model on ℕ 2 between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability 6 - 8 log 2 . When this happens, we also prove that the central cluster almost surely has a positive density on ℕ 2 . Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and on recent results about collision in the multi-TASEP.

How to cite

top

Coupier, David, and Heinrich, Philippe. "Coexistence probability in the last passage percolation model is $6-8\log 2$." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 973-988. <http://eudml.org/doc/271980>.

@article{Coupier2012,
abstract = {A competition model on $\mathbb \{N\}^\{2\}$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6-8\log 2$. When this happens, we also prove that the central cluster almost surely has a positive density on $\mathbb \{N\}^\{2\}$. Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and on recent results about collision in the multi-TASEP.},
author = {Coupier, David, Heinrich, Philippe},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {last passage percolation; totally asymmetric simple exclusion process; competition interface; second class particle; coupling},
language = {eng},
number = {4},
pages = {973-988},
publisher = {Gauthier-Villars},
title = {Coexistence probability in the last passage percolation model is $6-8\log 2$},
url = {http://eudml.org/doc/271980},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Coupier, David
AU - Heinrich, Philippe
TI - Coexistence probability in the last passage percolation model is $6-8\log 2$
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 973
EP - 988
AB - A competition model on $\mathbb {N}^{2}$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6-8\log 2$. When this happens, we also prove that the central cluster almost surely has a positive density on $\mathbb {N}^{2}$. Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and on recent results about collision in the multi-TASEP.
LA - eng
KW - last passage percolation; totally asymmetric simple exclusion process; competition interface; second class particle; coupling
UR - http://eudml.org/doc/271980
ER -

References

top
  1. [1] G. Amir, O. Angel and B. Valko. The tasep speed process. Available at arXiv:0811.3706, 2008. Zbl1225.82039MR2857238
  2. [2] D. Coupier and P. Heinrich. Stochastic domination for the last passage percolation model. Markov Process. Related Fields17 (2011) 37–48. Zbl1230.60102MR2809552
  3. [3] P. A. Ferrari, P. Gonçalves and J. B. Martin. Collision probabilities in the rarefaction fan of asymmetric exclusion processes. Ann. Inst. Henri Poincaré Probab. Stat.45 (2009) 1048–1064. Zbl1196.60162MR2572163
  4. [4] P. A. Ferrari, J. B. Martin and L. P. R. Pimentel. A phase transition for competition interfaces. Ann. Appl. Probab.19 (2009) 281–317. Zbl1185.60109MR2498679
  5. [5] P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab.33 (2005) 1235–1254. Zbl1078.60083MR2150188
  6. [6] O. Garet and R. Marchand. Coexistence in two-type first-passage percolation models. Ann. Appl. Probab.15 (2005) 298–330. Zbl1080.60092MR2115045
  7. [7] O. Häggström and R. Pemantle. First passage percolation and a model for competing spatial growth. J. Appl. Probab.35 (1998) 683–692. Zbl0920.60085MR1659548
  8. [8] C. Hoffman. Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab.15 (2005) 739–747. Zbl1067.60098MR2114988
  9. [9] C. Hoffman. Geodesics in first passage percolation. Ann. Appl. Probab.18 (2008) 1944–1969. Zbl1153.60055MR2462555
  10. [10] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer-Verlag, Berlin, 1999. Zbl0949.60006MR1717346
  11. [11] J. B. Martin. Last-passage percolation with general weight distribution. Markov Process. Related Fields12 (2006) 273–299. Zbl1147.60059MR2249632
  12. [12] T. Mountford and H. Guiol. The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab.15 (2005) 1227–1259. Zbl1069.60091
  13. [13] H. Rost. Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete58 (1981) 41–53. Zbl0451.60097
  14. [14] T. Seppäläinen. Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor K -exclusion processes. Trans. Amer. Math. Soc. 353 (2001) 4801–4829 (electronic). Zbl0983.60093
  15. [15] H. Thorisson. Coupling, Stationarity, and Regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. Zbl0949.60007

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.