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

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 t^2/3. 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].