Planar last passage percolation models are canonical examples of stochastic growth in the Kardar-Parisi-Zhang universality class, where one considers oriented paths between points in a random environment accruing the integral of the noise along itself as its weight. Given the endpoints, the extremal path with the maximum weight is termed as the geodesic.

As the endpoints are allowed to vary in space and time (one dimension for each), the joint ensemble of the weights gives rise to a four parameter space-time random energy field, whose conjectural universal weak limit, the Directed landscape, was recently constructed in a breakthrough work of Dauvergne, Ortmann and Virag. We shall discuss a few recent results identifying exponents that govern the space time geometry of this fundamental object and its prelimits.

In the first talk we shall study the aging behavior at short and large scales establishing exponents dictating the decay of correlations of weights in time, in last passage percolation on the lattice with exponential weights. We shall describe two results corresponding to the droplet and flat initial conditions confirming conjectures made by Ferrari and Spohn a few years ago.

In the second talk we shall describe results on fractal geometry specializing to a Brownian model. In particular, we study the coupling structure of the geodesic weight profiles at a fixed time (say 1) started at distinct points at time 0 by analyzing their difference function. Though in expectation this grows linearly, we show that the difference profile induces a random measure whose support is fractal and compute its dimension. We also relate the support of this measure to the exceptional set of points admitting disjoint geodesics.

Beyond geometric and probabilistic arguments involving geodesic behavior, the key inputs used are one point fluctuation information, locally Brownian nature of the geodesic weight profile, and sharp estimates on rarity of disjoint geodesics, the latter two being consequences of an invariance property under resampling termed as the Brownian Gibbs property.

These talks are based on a number of works jointly with subsets of Erik Bates, Alan Hammond, and Lingfu Zhang.