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The discrete and continuum broken line process. (English) Zbl 1200.60088

The authors introduce the discrete-space broken line process (with discrete and continuous parameter values) and derive some of its properties. It can be described as the following particle system with creation and annihilation. For space-time coordinates \((t,x)\in \mathbb{Z}^2\) with \(t+x\) even, the set of all such \((t,x)\) being denoted by \(\tilde{\mathbb{Z}}^2,\) consider independent random variables \(\xi_{t,x}\) which are geometrically distributed with parameters \(0<\lambda^2 <1: P(\xi_{t,x} =k)(1-\lambda^2)\lambda^{2k}, k=0,1,\dots\) At each time-space point \((t,x)\in \tilde{Z}^2, \xi_{t,x}\) pairs of particles are born with opposite velocities \(\pm 1.\) The particles move with their constant velocities and when moving particles with opposite velocities collide, they annihilate each other. The discrete broken line process consists of the space-time trajectories of the particles. The broken lines resemble the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented conditions that make the process stationary and self-dual. The authors also prove that the exponential and geometric distributions are the only non-trivial ones that yield self-duality.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60G60 Random fields
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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