Entropy-controlled Last-Passage Percolation

Abstract

We introduce 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 (path-entropy) constraint which takes into account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley’s LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. 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 (Berger and Torri (2018)). We give applications in this context, which are essentials tools in (Berger and Torri (2018)): we show that the limiting variational problem conjectured in (Ann. Probab. 44 (2016) 4006–4048), Conjecture 1.7, is finite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in (Comm. Pure Appl. Math. 64 (2011) 183–204; Probab. Theory Related Fields 137 (2007) 227–275).