Beyond Hammersley's Last-Passage Percolation: a discussion on possible local and global constraints

TL;DR

This paper extends Hammersley's Last-Passage Percolation model by introducing local and global constraints, analyzing how these affect the maximal path length in both directed and non-directed settings.

Contribution

It proposes a generalized LPP framework with H"older and path-entropy constraints, broadening the scope beyond traditional up-right paths and addressing non-directed cases.

Findings

Derived the order of the maximal path length under new constraints

Introduced H-LPP and E-LPP models with local and global conditions

Extended analysis to non-directed LPP scenarios

Abstract

Hammersley's Last-Passage Percolation (LPP), also known as Ulam's problem, is a well-studied model that can be described as follows: consider $m$ points chosen uniformly and independently in $[0,1]^2$, then what is the maximal number $\mathcal{L}_m$ of points that can be collected by an up-right path? We introduce here a generalization of this standard LPP, in order to allow for more general constraints than the up-right condition (a $1$-Lipschitz condition after rotation by $45^{\circ}$). We focus more specifically on two cases: (i) when the constraint is a $\gamma$-H\"older (local) condition, we call it H-LPP; (ii) when the constraint is a path-entropy (global) condition, we call it E-LPP. These generalizations also allows us to deal with non-directed LPP. We develop motivations for directed and non-directed constrained LPP, and we give the correct order of $\mathcal{L}_m$ in a general manner.