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

We show that continuous random walks (diffusion) in the Poincar\'{e} hyperbolic upper halfplane $\mathbb{H}^2 = \{(x,y)|y>0\}$ provide a unifying description of three seemingly unrelated phenomena: (i) the non-analytic divergence of the correlation length at the Berezinskii--Kosterlitz--Thouless (BKT) transition; (ii) the appearance of the Kardar--Parisi--Zhang (KPZ) exponent in the fluctuational behavior of stretched random walks constrained above an impermeable disc; and (iii) the emergence of Lifshitz tails (LT) in 1D statistics of rare events. We adapt the renormalization-group equations originally developed for the Efimov effect in a 2D conformally invariant potential to the case of diffusion in $\mathbb{H}^2$, thereby reproducing the BKT--type divergence of the correlation length. In frameworks of the same model we derive the KPZ--type behavior for the survival probability of stretched random walks near the boundary of $\mathbb{H}^2$ using scaling arguments, WKB--type approach, and numerical analysis. We demonstrate that LT emerge naturally in a deterministic large-deviation random walks' statistics in $\mathbb{H}^2$ via instanton approach, which rhymes with the rare-event behavior of 1D diffusion in the array of traps with the Poisson distribution. We conjecture that the dominant contribution to the statistics of paths responsible for BKT--like physics emerges from trajectories pushed to large-deviation stretched regime.