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

Abstract

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; (ii) the emergence of a Lifshitz tail (LT) in the 1D statistics of rare events; and (iii) the appearance of Kardar--Parisi--Zhang (KPZ) scaling in the fluctuations of stretched random walks constrained above an impermeable disc. We adapt the renormalization-group (RG) analysis originally developed for the Schrödinger equation with mixed boundary conditions in a 2D conformally invariant potential to the spectral problem in ${\mathbb{H}}^2$, and identify the conditions under which BKT-type behavior emerges. We show that the same RG equations can be interpreted in terms of the statistics of random walks in the large-deviation regime through an instanton approach, leading to Lifshitz tails (LT) in the associated spectral problem. We further derive KPZ-type scaling for the fluctuations of stretched random walks near the boundary of a disc in ${\mathbb{R}}^2$ and emphasize the similarity of this behavior with the boundary fluctuations of paths in ${\mathbb{H}}^2$ analyzed using a WKB approach.

31 pages, 4 figures, 2 appendices. The paper has been substantially revised