Discrete Hammersley's Lines with sources and sinks

TL;DR

This paper introduces two stationary discrete Hammersley's processes with sources and sinks, providing a unified approach to laws of large numbers in generalized Ulam's problems and characterizing extremal measures.

Contribution

It presents new stationary versions of discrete Hammersley's processes and offers a simplified, unified proof of laws of large numbers for generalized Ulam's problems.

Findings

Unified laws of large numbers for generalized Ulam's problems

Elementary solution for the original Ulam problem

Bernoulli product measures are the only extremal stationary measures on Z

Abstract

We introduce two stationary versions of two discrete variants of Hammersley's process in a finite box, this allows us to recover in a unified and simple way the laws of large numbers proved by T. Sepp{\"a}l{\"a}inen for two generalized Ulam's problems. As a by-product we obtain an elementary solution for the original Ulam problem. We also prove that for the first process defined on Z, Bernoulli product measures are the only extremal and translation-invariant stationary measures.