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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)].

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