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 H² = (x, y) |y>0, interpreted as multiplicative stochastic processes with log-normal statistics, provide a unifying framework linking three seemingly unrelated phenomena: (i) the non-analytic divergence of corrrelation length at the Berezinskii-Kosterlitz-Thouless (BKT) transition; (ii) the appearence of the Kardar-Parisi-Zhang 9KPZ) exponent in the fluctuational behavior of stretched random walks constrained above an impermeable disc; and (iii) the emergence of Lifshitz tails in one-dimensional statistics of rare events. Combining scaling arguments with analytic derivations and numerical analysis, we adapt the renormalization-group equations originally developed for the Efimov effect in a two-dimensional conformally invariant potential to the case of diffusion in H², thereby deriving the BKT-type divergence of the correlation length. We further demonstrate how the KPZ-type scaling governs the large-deviation behavior and survival probability near the boundary in the hyperbolic domain, and how Lifshitz tails arise naturally in a deterministic large-deviation landscape on the hyperbolic plane via instanton approach, reproducing the rare-event statistics of one-dimensional diffusion in the array of traps with the Poisson distribution. We conjecture that the dominant contribution to the ensemble of paths responsible for BKT-like physics comes from random paths pushed to large-deviation stretched regime.