DIRECTED AND NON-DIRECTED PATH CONSTRAINED LAST-PASSAGE PERCOLATION

Quentin Berger (1) , Torri Niccolo (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 comes from the $\gamma$-H\"older norm of the path (a local condition), we call it H-LPP; (ii) when the constraint comes from the entropy of a path (a global condition), we call it E-LPP. These generalizations of the standard LPP also allows us to deal with \textit{non-directed} LPP. We develop motivations for directed and non-directed path-constrained LPP, and we find the correct order of $\mathcal{L}_m$ in a general manner -- as a specific example, the maximal number of points that can be collected by a non-directed path of total length smaller than~$1$ is shown to be of order $\sqrt{m}$. This new LPP opens the way for many interesting problems, and we present some of its potential applications, to the context of directed and non-directed polymers in random environment.

Mots clés: Last-passage percolation; directed polymers; non-directed polymers; Hammersley's Process

Domaines: Mathématiques [math]; Probabilités [math.PR]

Quentin Berger, Torri Niccolo. DIRECTED AND NON-DIRECTED PATH CONSTRAINED LAST-PASSAGE PERCOLATION. 2018. ⟨hal-01706671v2⟩