Entropy-controlled Last-Passage Percolation

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] (Conjecture 1.7) 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].