Abstract:

This thesis focuses on the study of directed last passage percolation (LPP) models in random media, which are two-dimensional stochastic growth models. These models have numerous applications and exhibit deep connections with random matrix theory and integrable probability. After appropriate scaling, it has been shown that LPP models converge to a universal limit known as the KPZ fixed point. A key quantity in these models is the last passage time, which represents the maximum total weight collected along an up-right path between two points. The thesis is divided into two parts.

The first part of this thesis studies exponential LPP under the condition that the last passage time at one point is unusually large. We then analyze the effect of this conditioning on the last passage time at other points. We find that this conditioning influences the last passage times in certain regions. After applying a suitable scaling, which differs from the 1 : 2 : 3 KPZ scaling, we also derive results for the conditional fluctuations.

The second part of the thesis is devoted to computing the multi-point distributions for Poissonian and Brownian LPP with step initial condition. Using these explicit formulas, we provide an alternative proof that the multi-point distributions of Poissonian and Brownian LPP converge to those of the KPZ fixed point under the 1 : 2 : 3 scaling.