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Article Dans Une Revue Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions Année : 2021

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

Résumé

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 $γ$-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.
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Dates et versions

hal-01706671 , version 1 (12-02-2018)
hal-01706671 , version 2 (05-04-2018)
hal-01706671 , version 3 (08-06-2018)

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  • HAL Id : hal-01706671 , version 3

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Quentin Berger, Torri Niccolo. Beyond Hammersley's Last-Passage Percolation: a discussion on possible local and global constraints. Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, 2021, 8 (2), pp.213-241. ⟨hal-01706671v3⟩
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