Entropy-controlled Last-Passage Percolation

Abstract: In the present article we consider a natural generalization of Hammersley's Last Passage Percolation (LPP) called Entropy-controlled Last Passage Percolation (E-LPP), where points can be collected by paths with a global (entropy) constraint which takes in account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley's LPP. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in the related paper [Berger and Torri, 2018]. We prove several estimates on the E-LPP in continuous and in discrete settings, which are of interest on their own. We give applications in the context of polymers in heavy-tail environment which are essentials tools in [Berger and Torri, 2018]: we show that the limiting variational problem conjectured by Dey and Zygouras, 2016 is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used by [Auffinger-Louidor, 2011] and [Hambly and Martin, 2007].