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 2E(t − X(t))+, where X(t) is the location of a second class particle at time t. This implies that both 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].

Busemann functions and equilibrium measures in last passage percolation models

“…This does seem to be the most important contribution of the interacting fluid representation: once we have a good candidate for ν α , we can check it by showing that it is invariant under the evolution of the interacting fluid. In fact, even in the results for the classical Hammersley case found in Aldous and Diaconis [1] and Cator and Groeneboom [5,6], this is where the interacting particle process proves its worth.…”

“…Furthermore, we conjecture that under suitable conditions, the random measures ν α behave asymptotically like a Brownian motion plus linear drift (just like a compound Poisson process). If this is true, we believe that we can extend the methods from [6] to conclude cube-root asymptotics for the length of the longest paths and its fluctuation. This idea will be pursued in an upcoming paper.…”

“…In the classical model, the exit point formula for the equilibrium regime, proved by Cator and Groeneboom [6], is…”

“…It only relies on an explicit calculation of the generator associated to M t . (See Lemma 8 of [1], or Theorem 3.1 of [6]). …”

Weakly Asymmetric Exclusion and KPZ

“…We now give a brief hint at the proof of the key estimate (38). It is adapted from earlier work of [5], which in turn goes back to [11].…”