Kardar, Parisi and Zhang proposed a model to study universal fluctuation behaviour occurring in a wide class of random growth models, now known as the KPZ universality class. In this thesis we will focus on some planar random growth models 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.

In this talk we focus on the second part. We consider sequences of last passage times, along diagonal and vertical directions. Under the KPZ fluctuation, the weak convergence of these sequences is known, and one natural question is the a.s. order of limsup and liminf of these sequences. We obtain laws of iterated logarithm (LIL) and laws of fractional logarithm (LFL) for diagonal and vertical directions respectively. Owing to the connections between LPP and random matrix theory, these results resolve some conjectures in the latter setting.