2018/02/12 by Quentin Berger, Berger, Quentin, Niccolò Torri +1 · 1 citation
Mathematics · Physics and Astronomy · #60K35 #82B44 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1802.04046
openalex publication_date 2018/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hammersley's Last-Passage Percolation (LPP), also known as Ulam's problem, is\na well-studied model that can be described as follows: consider m points\nchosen uniformly and independently in [0,1]2, then what is the maximal\nnumber \Lm of points that can be collected by an up-right path? We\nintroduce here a generalization of this standard LPP, in order to allow for\nmore general constraints than the up-right condition (a 1-Lipschitz condition\nafter rotation by 45\∘). We focus more specifically on two cases: (i)\nwhen the constraint is a \γ-H "older (local) condition, we call it H-LPP;\n(ii) when the constraint is a path-entropy (global) condition, we call it\nE-LPP. These generalizations also allows us to deal with non-directed LPP. We\ndevelop motivations for directed and non-directed constrained LPP, and we give\nthe correct order of \Lm in a general manner.\n