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.

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

References in corpus (13)

• The one-dimensional KPZ equation and its universality class
• Second class particles and cube root asymptotics for Hammersley's process
• Competition interfaces and second class particles
• The strict-weak lattice polymer
• The motion of a second class particle for the tasep starting from a decreasing shock profile
• A phase transition for competition interfaces
• Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
• Hammersley's process with sources and sinks
• Local stationarity of exponential last passage percolation
• Right-tail moderate deviations in the exponential last-passage percolation
• Moderate deviation and exit time estimates for stationary Last Passage Percolation
• On the exponent governing the correlation decay of the Airy${}_1$ process
• Central moments of the free energy of the O'Connell-Yor polymer

Cited by in corpus (5)

• Large deviations for the $q$-deformed polynuclear growth
• Tail estimates for the stationary stochastic six vertex model and ASEP
• Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation
• The stochastic six-vertex model speed process
• Geodesic trees in last passage percolation and some related problems