The Discrete and Continuum Broken Line Process
2010, v.16, Issue 1, 79-116
In this work we introduce the discrete-space broken line process (with discrete and continuous parameter values) and derive some of its properties. We explore polygonal Markov field techniques developed by Arak - Surgailis. The discrete version is presented first and a natural generalization to a continuous object living on the discrete lattice is then proposed and studied The broken lines also resemble the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and self-dual. For such distributions there is a law of large numbers and the process extends to the infinite lattice. A proof of Burke's theorem emerges from the construction. We present a simple proof of the explicit law of large numbers for last passage percolation as an application. Finally we show that the exponential and geometric distributions are the only non-trivial ones that yield self-duality.
Keywords: spatial random processes,Hammersley process,last passagepercolation,time constant,broken line process