Optimal tail exponents in general last passage percolation via bootstrapping & geodesic geometry

We consider last passage percolation on Z(2) with general weight distributions, which is expected to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. In this model, an oriented path between given endpoints which maximizes the sum of the i.i.d. weight variables associated to its vertices is called a geodesic. Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the tail assumptions to prove optimal upper and lower tail behavior with the exponents of 3/2 and 3 for the weight of the geodesic from (1, 1) to (r, r) for all large finite r, and thus unearth a connection between the tail exponents and the characteristic KPZ weight fluctuation exponent of 1/3. The proofs merge several ideas which are not reliant on the exact form of the vertex weight distribution, including the well known super-additivity property of last passage values, concentration of measure behavior for sums of stretched exponential random variables, and geometric insights coming from the study of geodesics and more general objects called geodesic watermelons. Previous proofs of such optimal estimates have relied on hard analysis of precise formulas available only in integrable models. Our results illustrate a facet of universality in a class of KPZ stochastic growth models and provide a geometric explanation of the upper and lower tail exponents of the GUE Tracy-Widom distribution, the conjectured one point scaling limit of such models. The key arguments are based on an observation of general interest that super-additivity allows a natural iterative bootstrapping procedure to obtain improved tail estimates.