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

Cator, Eric; Groeneboom, Piet

Ann. Probab. 34, No. 4, 1273-1295 (2006).

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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