  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.
