Discrete Gibbsian line ensembles and weak noise scaling for directed polymers

In this thesis we investigate three projects within in the field of KPZ universality class and integrable probability. The first project studies the weak KPZ universality for half-space directed polymers in dimension 1+1. This is the half-space analogue of the full-space polymers studied in [AKQ]. The novelty is the extra random environment introduced at the boundary. The new technical challenges are the estimates for half-space heat kernels which are super-probability measures and accurate estimates on visits to origin (weighted by the boundary randomness) for a simple symmetric walk in dimension 1 and 2 respectively. The second project introduces a framework to prove tightness of a sequence of discrete Gibbsian line ensembles L^N = {L(N/k)(u), k ∈ N, u ∈ (1/N)Z}, which is a countable collection of random curves. The sequence of discrete line ensembles L^N we consider enjoys a resampling invariance property, which we call (H^N,H^R^W,^N )-Gibbs property. We assume that L^N satisfies technical assumptions A1-A4 on (H^N,H^R^W,^N ) and the assumption that the lowest labeled curve with a parabolic shift, L(N/1)(u)+ u²/2 , converges weakly to a stationary process in the topology of unif