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 uniform convergence on compact sets. Under these assumptions, we prove our main result Theorem 3.1.13 that L^N is tight as a sequence of line ensembles and that the H-Brownian Gibbs property holds for all subsequential limit line ensembles with H(x) = e^x. As an application of Theorem 3.1.13, under the weak noise scaling, we show that the scaled log-gamma line ensemble L^N is tight, which is a sequence of discrete line ensembles associated with the log-gamma polymer model via the geometric RSK correspondence. The H-Brownian Gibbs property (with H(x) = e^x) of its subsequential limits also follows. The third project proves an analogue of the classical Komlós-Major-Tusnády (KMT) embedding theorem for random walk bridges and it serves as a key technical input for the second project. The random bridges we consider are constructed through random walks with i.i.d jumps that are conditioned on the locations of their endpoints. We prove that such bridges can be strongly coupled to Brownian bridges of appropriate variance when the jumps are either continuous or integer valued under some mild technical assumptions on the jump distributions. Our arguments follow a similar dyadic scheme to KMT's original proof, but they require more refined estimates and stronger assumptions necessitated by the endpoint conditioning. In particular, our result does not follow from the KMT embedding theorem, which we illustrate via a counterexample.