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.

Non-existence of non-trivial bi-infinite geodesics in Geometric Last Passage Percolation

“…Along the way, we also prove some novel results about geometric last-passage percolation in order to verify our hypotheses. In particular, we prove a new, sharp bound for exit times in increment-stationary geometric last-passage percolation following a strategy recently introduced in [21,22]. This is recorded as Theorem B.1 below.…”

Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation

“…More specifically, this article presents a new derivation of optimal-order central moment bounds in planar directed last-passage percolation (LPP). Our technique builds on a recent strengthening [49,50] of the coupling method, and requires no input from integrable probability or random matrix theory. A main appeal of our approach is that it would in principle be conveniently applicable in both zero and positive temperature settings, as long as the associated increment-stationary LPP and free energy processes are sufficiently tractable.…”

“…The present work introduces our argument in the context of the exponential LPP, one of the quintessential models in the KPZ universality class, with the aim of serving as a detailed blueprint for potential adaptations to the aforementioned settings. Because the inputs from [49,50] were developed for the exponential LPP, this model is also where our presentation can be most concise. 1.2.…”

“…The follow-up work of the authors [93] tailored their technique to the four basic integrable lattice polymers using the framework of [31]. Around the same time, articles [49,50] made the following observations. Certain m.g.f.…”

“…These subsume the variance identities and elevate the capabilities of the coupling method to the level of integrable probability for many purposes. For example, article [50] derived optimal-order geodesic fluctuation bounds through the enhanced coupling method. Among the several other applications announced in [50], one concerns central moment bounds, which is the topic of the present work.…”