Short Talk: Kaihao Jing

Last passage percolation (LPP) is a model of random geometry where the main observable is a path evolving in a random environment. When the environment distribution has light tails and a fast decay of correlation, the random fluctuations of LPP are predicted to be explained by the Kardar–Parisi–Zhang (KPZ) universality theory. However, in strongly correlated environments — such as those arising from branching random walks, Gaussian free field and other log-correlated fields— KPZ predictions are believed to break down, and much less is known. In this talk we will present recent progress towards developing an understanding of LPP in such environments. In particular, we will report results on LPP in fractal percolation, a random fractal set introduced by Mandelbrot, a hierarchical approximation of the “thick points” of log-correlated fields. We show that in this setting, the LPP energy grows sub-linearly — in contrast to the linear growth under KPZ universality. I will also discuss results in high dimension which provides a rich setting to investigate the structural distinction between directed and undirected models of random geometry. This is based on joint work with Shirshendu Ganguly and Victor Ginsburg.