The Discrete and Continuum Broken Line Process

TL;DR

This paper introduces a novel discrete and continuum broken line process, exploring its properties, connections to known models, and applications in last passage percolation, including stationarity, self-duality, and explicit law of large numbers.

Contribution

It develops a new broken line process model, generalizes it to the continuum, and establishes key properties like stationarity, self-duality, and explicit limit laws.

Findings

The process is stationary and self-dual for certain distributions.

Explicit law of large numbers for last passage percolation is proved.

Exponential and geometric distributions uniquely yield self-duality.

Abstract

In this work we introduce the discrete-space broken line process (with discrete and continues parameter values) and derive some of its properties. We explore polygonal Markov fields techniques developed by Arak-Surgailis. The discrete version is presented first and a natural continuum 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.