Discrete geodesic local time converges under KPZ scaling

Abstract: The directed landscape constructed in (Dauvergne-Ortmann-Virag '18) produces a directed, planar, random geometry, and is believed to be the universal scaling limit of two-dimensional first and last passage percolation models in the Kardar-Parisi-Zhang (KPZ) universality class. Geodesics in this random geometry form an important class of random continuous curves exhibiting fluctuation theory quite different from that of Brownian motion. In this vein, counterpart to Brownian local time, BLT (a self-similar measure supported on the set of zeros of Brownian motion), a local time for geodesics, GLT, was recently constructed and used to study fractal properties of the directed landscape in (Ganguly-Zhang '22). It is a classical fact and can be proven using the Markovian property of Brownian motion that the uniform discrete measure on the set of zeros of the simple random walk converges to BLT. In this paper, we prove the ''KPZ analog'' of this by showing that the local times for discrete geodesics in pre-limiting integrable last passage percolation models converge to GLT under suitable scaling guided by KPZ exponents. In absence of any Markovianity, our arguments rely on the recently proven convergence of geodesics in pre-limiting models to that in the directed landscape (Dauvergne-Virag '21). However, this input concerns macroscopic properties and is too coarse to capture the microscopic information required for local time analysis. To relate the macroscopic and microscopic behavior, a key ingredient is an a priori smoothness estimate of the local time in the discrete model, proved relying on geometric ideas such as the coalescence of geodesics as well as their stability under perturbations of boundary data.