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

TL;DR

The paper develops a unifying framework for diffusion in the Poincaré upper half-plane $\mathbb{H}^2$, showing that multiplicative, log-normal fluctuations in hyperbolic geometry connect three seemingly distinct phenomena: non-analytic BKT divergence of the correlation length, KPZ-like fluctuations of stretched paths near boundaries, and Lifshitz-tail statistics in large-deviation regimes. By mapping diffusion in $\mathbb{H}^2$ to a 2D conformally invariant radial problem and employing an Efimov/RG approach, it derives a BKT-type exponential divergence of the correlation length and interprets the associated energy gap as a Lifshitz tail, even in a disorder-free setting. A deterministic large-deviation landscape underlies this connection, with a cubic RG potential governing the flow and yielding instanton actions that reproduce Lifshitz-tail scalings. In the boundary-constraint problems, stretched diffusion exhibits KPZ-like $1/3$ scaling, demonstrated via Euclidean Airy-squared statistics and a WKB analysis in $\mathbb{H}^2$, establishing a unified picture where stretched hyperbolic trajectories dominate the long-time behavior. The results offer a geometric, cross-domain perspective on complex fluctuation phenomena and suggest a broad applicability to other curved-space diffusion problems and large-deviation processes.

Abstract

We show that continuous random walks (diffusion) in the Poincaré 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.

Figures (4)

• Figure 1: Flowchart: Three faces of random walks in $\mathbb{H}^2$.
• Figure 2: (a) Brownian bridge above the disc of radius $R$ in $\mathbb{R}^2$ stretched along boundary, where $\Delta r_{\mathbb{R}}$ and $l_{\mathbb{R}}$, are the typical "span" and the "correlation lengths" of the stretched Brownian bridge; (b) Schematic picture of Brownian bridge in the Poincaré domain $\mathbb{H}^2$ in WKB approximation, which corresponds to the bridge in panel (a), where $\Delta y_{\mathbb{H}}$ and $\Delta x_{\mathbb{H}}$ match $\Delta r_{\mathbb{R}}$ and $l_{\mathbb{R}}$.
• Figure 3: Left: Expectation $\Delta r_{\mathbb{E}}(R)$ as a function of $R$ for stretched paths of length $L\equiv t = c R$ in double-logarithmic coordinates; Right: Comparison of the distribution $\Omega(r)$ with $\mathrm{const}\; \mathrm{Ai}^2(\beta R^{-1/3}r+\tilde{a}_1)$ for $R=100$, $c=10$ and mixed (third-type) boundary conditions with $\kappa=0.2$, $\tilde{a}_1\approx -1.953$, $\beta\approx 0.357$
• Figure 4: Left: Expectation $\Delta r_{\mathbb{E}}(R)$ as a function of $R$ for stretched paths of length $L\equiv t = c R$ in double-logarithmic coordinates; Right: Comparison of the distribution $\Omega(r)$ with $\mathrm{const}\; \mathrm{Ai}^2(\beta R^{-1/3}r+a_1)$ for $R=100$, $c=10$ and Dirichlet boundary conditions. $\beta\approx 0.371$