  We consider last passage percolation on $\mathbb Z^2$ 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
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 $r$. The proofs merge several ideas, 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. Previously such optimal behavior was only known for
exactly solvable models, with proofs relying on hard analysis of formulas from
integrable probability, which are unavailable in the general setting. 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.
