In 1986 Kardar, Parisi and Zhang proposed a stochastic PDE to study universal fluctuation behaviour occurring in a wide class of random growth models. This class is now known as the KPZ universality class. In this thesis we will focus on some planar random growth models that have been proved to be in the KPZ class. In particular, we will study three last passage percolation (LPP) models: exponential, Brownian and Poissonian LPP. These models can be viewed as random geometric objects, and due to exact formulas, a detailed analysis of these models is possible. The thesis can be divided into two parts. In the first part we study the geodesic tree in exponential LPP. In the second part, we develop geometric techniques in various LPP models, to establish some limit theorems and related results.

One interesting behaviour the above random geometric models show is the coalescence of geodesics. For example, in exponential LPP on $\mathbb{Z}^2$, it is known that, for a fixed (non-axial) direction, almost surely, semi-infinite geodesics in that direction coalesce and form a one-ended tree. In this talk, we will address several natural questions associated with these geodesic trees. In particular, we obtain optimal (up to constants) upper and lower bounds for the tails of the height and the volume of the backward sub-tree rooted at a fixed point. We also obtain bounds for the probability that the sub-tree in the direction (1,1) rooted at the origin contains the vertex $-(n,n)$, which answers a question analogous to the well-known midpoint problem in the context of semi-infinite geodesics. Furthermore, we obtain bounds for the probability that a pair of intersecting geodesics both pass through a given vertex.