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

We show that continuous random walks (diffusion) in the Poincaré hyperbolic upper halfplane ${\mathbb{H}}^2=\{(x,y)\mathrm{\mid }y>0\}$ 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.