This talk is about limit theorem for increments of Brownian local time. In a first part, we will look at limit theorems for moments of such increments. As special cases for the second and third moments, previous results by Chen et al. and Rosen, which were later re-proven by Hu and Nualart and Rosen are included and a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a different approach by exclusively working in the space variable of the Brownian local time, which allows us to give a unified argument for arbitrary orders. In a second part, still working in the space variable but using a different argument, some closely related results for generic smooth transformations of Brownian local time increments will be presented.

We introduce a class of 1+1 growth processes that are new to the literature, which we call "doubly geometric". These models are defined via the Higher Spin Six-Vertex Model and analyzed through determinantal formulas from Schur processes. We give the limit shape for these processes and show that the fluctuations around the limit shape are given by the Tracy-Widom distribution. These results are similar to other models in the Kardar-Parisi-Zhang universality class. This is joint work with Leo Petrov (UVa) and Alisa Knizel (Columbia).