Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
Abstract
We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are of optimal cubic-exponential order. We derive them in the context of last-passage percolation with exponential weights with near-stationary boundary conditions. As a result, the probabilistic coupling method is empowered to treat a variety of problems optimally, which could previously be achieved only via inputs from integrable probability. As applications in the bulk setting, we obtain upper bounds of cubic-exponential order for transversal fluctuations of geodesics, and cube-root upper bounds with a logarithmic correction for distributional Busemann limits and competition interface limits. Several other applications are already in the literature.
Cite
@article{arxiv.2105.09402,
title = {Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique},
author = {Elnur Emrah and Christopher Janjigian and Timo Seppäläinen},
journal= {arXiv preprint arXiv:2105.09402},
year = {2023}
}
Comments
v3: Added MSP copyright notice. v2: Accepted version. Two main corrections: Added to Prop. 3.6 a missing assumption that was used in the proof. (The z-parameter must be close to zeta). Also added a related assumption to Thm. 4.4(b) (previously Prop. C.1), which relies on Prop. 3.6. Further minor corrections and edits. Slightly improved Thm. 3.4 and Prop. B.6. Significantly revised introduction