2020/07/07 by Shirshendu Ganguly, Ganguly, Shirshendu, Milind Hegde +1 · 1 citation
Mathematics · #Combinatorics #Exponent #Exponential function #FOS: Mathematics #FOS: Physical sciences #Geodesic #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Probability (math.PR) #Random Matrices and Applications #Renormalization group #Statistical Methods and Bayesian Inference #Statistical physics #Stochastic processes and statistical mechanics #Universality (dynamical systems)
paper · pdf · doi:10.48550/arxiv.2007.03594
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2020/07/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/06
We consider last passage percolation on \mathbb Z2 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.