Beyond Hammersley's last-passage percolation: a discussion on possible local and global constraints (Q2031488)

Summary: Hammersley's last-passage percolation (LPP), also known as Ulam's problem, is a well-studied model that can be described as follows: let \(m\) points be 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 LPP, allowing for more general constraints than the up-right condition: the constraints may be either \textit{local} or \textit{global}. We give the correct order of \(\mathcal{L}_m\) in a general manner, and we illustrate the interest and usefulness of this generalized LPP with examples and simulations.