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

Discrete geodesic local time converges under KPZ scaling