2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/110190We 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].Published at http://dx.doi.org/10.1214/009117906000000089 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)ProbabilityMathematical Physics60C05, 60K35 (Primary) 60F05 (Secondary)text