A shape theorem and semi-infinite geodesics for the Hammersley model with random weights

TL;DR

This paper establishes a shape theorem and fluctuation bounds for the Hammersley last passage percolation model with random weights, and discusses implications for semi-infinite geodesics and future research directions.

Contribution

It proves a shape theorem with fluctuation bounds for the Hammersley model with random weights and explores semi-infinite geodesics, extending previous work to more general settings.

Findings

Shape theorem for the Hammersley model established.

Diffusive upper bounds for shape fluctuations derived.

Existence and coalescence of semi-infinite geodesics discussed.

Abstract

In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these results can be used to prove existence and coalescence of semi-infinite geodesics in some fixed direction $\alpha$, following an approach developed by Newman and co-authors, and applied to the classical Hammersley process by W\"uthrich. These results will be crucial in the development of an upcoming paper on the relation between Busemann functions and equilibrium measures in last passage percolation models.