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Working in the setting of i.i.d. last-passage percolation on $\mathbb{R}^D$with no assumptions on the underlying edge\hyp{}weight distribution, we arriveat the notion of grid entropy - a Subadditive Ergodic Theorem limit of theentropies of paths with empirical measures weakly converging to a given target,or equivalently a deterministic critical exponent of canonical order statisticsassociated with the Levy-Prokhorov metric. This provides a fresh approach to anentropy first developed by Rassoul-Agha and Sepp\"al\"ainen as a largedeviation rate function of empirical measures along paths. In their 2014 paperarXiv:1202.2584, variational formulas are developed for thepoint-to-point/point-to-level Gibbs Free Energies as the convex conjugates ofthis entropy. We rework these formulas in our new framework and explicitly linkour descriptions of grid entropy to theirs. We also improve on a known boundfor this entropy by introducing a relative entropy term in the inequality.Furthermore, we show that the set of measures with finite grid entropycoincides with the deterministic set of limit points of empirical measuresstudied in a recent paper arXiv:2006.12580 by Bates. In addition, we partiallyanswer a directed polymer version of a question of Hoffman which was previouslytackled in the zero temperature case by Bates. Our results cover both thepoint-to-point and point-to-level scenarios.

Grid entropy in last passage percolation -- a superadditive critical exponent approach