Three faces of random walks in a hyperbolic domain: BKT, Lifshitz tails, and KPZ

We show that continuous random walks (diffusion) in the Poincaré upper half-plane $\mathbb{H}^2 = \{(x,y)| y>0\}$ provide a unified description of three seemingly unrelated phenomena: (i) the non-analytic divergence of the correlation length at the Berezinskii--Kosterlitz--Thouless (BKT) transition;