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

TL;DR

This paper reveals how random walks in hyperbolic space unify phenomena like BKT transition, KPZ scaling, and Lifshitz tails through analytic, scaling, and numerical methods, highlighting their interconnected nature.

Contribution

It introduces a unified framework linking BKT, KPZ, and Lifshitz tails via hyperbolic diffusion, adapting RG equations and large-deviation analysis.

Findings

Derives BKT divergence of correlation length in hyperbolic diffusion.

Shows KPZ scaling governs large-deviation behavior near boundaries.

Reproduces Lifshitz tails in rare-event statistics using instanton methods.

Abstract

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