Busemann functions and the speed of a second class particle in the rarefaction fan. (English) Zbl 1276.60108

Summary: We show how the results in [the authors, Probab. Theory Relat. Fields 154, No. 1–2, 89–125 (2012; Zbl 1262.60094)], about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymmetric exclusion process or the Hammersley interacting particle process. The method is to use the well-known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well-known connection between second class particles and competition interfaces.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics


Zbl 1262.60094
Full Text: DOI arXiv Euclid


[1] Aldous, D. and Diaconis, P. (1995). Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199-213. · Zbl 0836.60107
[2] Amir, G., Angel, O. and Valkó, B. (2011). The TASEP speed process. Ann. Probab. 39 1205-1242. · Zbl 1225.82039
[3] 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). · Zbl 1139.60046
[4] Cator, E. and Dobrynin, S. (2006). Behavior of a second class particle in Hammersley’s process. Electron. J. Probab. 11 670-685 (electronic). · Zbl 1112.60076
[5] Cator, E. and Groeneboom, P. (2005). Hammersley’s process with sources and sinks. Ann. Probab. 33 879-903. · Zbl 1066.60011
[6] Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 1273-1295. · Zbl 1101.60076
[7] Cator, E. A. and Pimentel, L. P. R. (2012). Busemann functions and equilibrium measures in last-passage percolation. Probab. Theory Related Fields 154 89-125. · Zbl 1262.60094
[8] Cohen, J. W. (1969). The Single Server Queue. North-Holland Series in Applied Mathematics and Mechanics 8 . North-Holland, Amsterdam. · Zbl 0183.49204
[9] Coletti, C. F. and Pimentel, L. P. R. (2007). On the collision between two PNG droplets. J. Stat. Phys. 126 1145-1164. · Zbl 1120.82015
[10] Coupier, D. (2011). Multiple geodesics with the same direction. Available at . · Zbl 1244.60093
[11] Ferrari, P. A. and Kipnis, C. (1995). Second class particles in the rarefaction fan. Ann. Inst. Henri Poincaré Probab. Stat. 31 143-154. · Zbl 0813.60095
[12] Ferrari, P. A., Martin, J. B. and Pimentel, L. P. R. (2009). A phase transition for competition interfaces. Ann. Appl. Probab. 19 281-317. · Zbl 1185.60109
[13] Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Probab. 33 1235-1254. · Zbl 1078.60083
[14] 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. · Zbl 1069.60091
[15] 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. · Zbl 0848.60089
[16] Pimentel, L. P. R. (2007). Multitype shape theorems for first passage percolation models. Adv. in Appl. Probab. 39 53-76. · Zbl 1112.60083
[17] Pyke, R. (1959). The supremum and infimum of the Poisson process. Ann. Math. Statist. 30 568-576. · Zbl 0089.13602
[18] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41-53. · Zbl 0451.60097
[19] Seppäläinen, T. (1998). Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields 4 1-26. · Zbl 0906.60082
[20] 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, MA. · Zbl 1011.60085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.