Annals of Probability

Second class particles and cube root asymptotics for Hammersley’s process

Eric Cator and Piet Groeneboom

Full-text: Open access

Abstract

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 t2/3. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879–903].

Article information

Source
Ann. Probab., Volume 34, Number 4 (2006), 1273-1295.

Dates
First available in Project Euclid: 19 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1158673319

Digital Object Identifier
doi:10.1214/009117906000000089

Mathematical Reviews number (MathSciNet)
MR2257647

Zentralblatt MATH identifier
1101.60076

Subjects
Primary: 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems

Keywords
Longest increasing subsequence Ulam’s problem Hammersley’s process cube root convergence second class particles Burke’s theorem

Citation

Cator, Eric; Groeneboom, Piet. Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 (2006), no. 4, 1273--1295. doi:10.1214/009117906000000089. https://projecteuclid.org/euclid.aop/1158673319


Export citation

References

  • Aldous, D. and Diaconis, P. (1995). Hammersley's interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199–213.
  • 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.
  • Baik, J. and Rains, E. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Statist. Phys. 100 523–541.
  • Cator, E. and Groeneboom, P. (2005). Hammersley's process with sources and sinks. Ann. Probab. 33 879–903.
  • Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 820–832.
  • Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79–109.
  • Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445–456.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
  • 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.