Second class particles and cube root asymptotics for Hammersley’s process. (English) Zbl 1101.60076

Summary: We show that, for a stationary version of Hammersley’s process, with Poisson sources on the positive \(x\)-axis and Poisson sinks on the positive \(y\)-axis, the variance of the length of a longest weakly North-East path \(L(t,t)\) from \((0,0)\) to \((t,t)\) is equal to \(2\mathbb E(t-X(t))_{+}\), where \(X(t)\) is the location of a second class particle at time \(t\). This implies that both \(\mathbb E(t-X(t))_{+}\) and the variance of \(L(t,t)\) are of order \(t^{2/3}\). Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of the authors [Ann. Probab. 33, 879–903 (2005; Zbl 1066.60011)].


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60C05 Combinatorial probability


Zbl 1066.60011
Full Text: DOI arXiv


[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] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequences of random permutations. J. Amer. Math. Soc. 12 1119–1178. JSTOR: · Zbl 0932.05001
[3] Baik, J. and Rains, E. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Statist. Phys. 100 523–541. · Zbl 0976.82043
[4] Cator, E. and Groeneboom, P. (2005). Hammersley’s process with sources and sinks. Ann. Probab. 33 879–903. · Zbl 1066.60011
[5] Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 820–832. · Zbl 0806.60099
[6] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79–109.
[7] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445–456. · Zbl 0960.60097
[8] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219. · Zbl 0703.62063
[9] Seppäläinen, T. (2005). Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33 759–797. · Zbl 1108.60083
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.