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arxiv: 2511.18510 · v2 · pith:XBWH5GO4new · submitted 2025-11-23 · ❄️ cond-mat.stat-mech · hep-th

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

Daniil Fedotov , Sergei Nechaev This is my paper

Pith reviewed 2026-05-25 07:52 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-th
keywords random walkshyperbolic geometryBKT transitionKPZ equationLifshitz tailsEfimov effectlarge deviations
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The pith

Diffusion in the hyperbolic plane unifies the BKT transition, KPZ fluctuations, and Lifshitz tails.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that continuous random walks in the Poincaré upper half-plane H² serve as a single geometric setting that reproduces three distinct statistical phenomena. It adapts renormalization-group flow equations originally derived for the Efimov effect to recover the essential singularity in the correlation length at the Berezinskii–Kosterlitz–Thouless transition. The same diffusion process is then shown to generate the KPZ roughness exponent for the survival probability of walks stretched above an impermeable boundary, and to produce Lifshitz tails in the large-deviation statistics of rare paths via an instanton calculation. The authors conjecture that the dominant trajectories responsible for BKT physics are those that enter the stretched regime.

Core claim

Continuous random walks in H² = {(x,y) | y > 0} simultaneously encode the non-analytic divergence of the correlation length at the BKT transition, the KPZ exponent in the survival probability of stretched walks near the boundary, and the Lifshitz tails that appear in the deterministic large-deviation statistics of paths; the same adapted RG equations that capture the BKT singularity also allow WKB and numerical extraction of the KPZ scaling and instanton derivation of the Lifshitz tails.

What carries the argument

Continuous diffusion (random walks) in the Poincaré hyperbolic upper half-plane H², with the adapted renormalization-group flow from the Efimov problem supplying the BKT divergence and scaling arguments plus instantons supplying the KPZ and Lifshitz behaviors.

If this is right

  • The BKT correlation length diverges as exp(c / sqrt(T - Tc)) when the RG flow for diffusion in H² is solved.
  • The survival probability of walks constrained above the boundary of H² decays with the KPZ roughness exponent 1/3 in the stretched regime.
  • The probability of rare large-deviation paths in H² develops Lifshitz tails whose functional form matches the 1D Poisson-trap problem.
  • BKT-like physics is conjectured to be carried by the subset of trajectories that reach the large-deviation stretched regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unification suggests that hyperbolic geometry may underlie other BKT–KPZ crossovers in planar systems without explicit curvature.
  • The instanton treatment of Lifshitz tails could be tested by measuring the tail of the distribution of maximal heights of random walks on a disc.
  • If the conjecture holds, finite-size scaling near the BKT point should show a crossover from ordinary diffusive to stretched-path dominance at a length scale set by the hyperbolic metric.

Load-bearing premise

The renormalization-group equations developed for the Efimov effect in a 2D conformally invariant potential can be transplanted to diffusion in H² without modifications that would change the form of the BKT divergence.

What would settle it

Direct numerical integration of the diffusion equation or Monte Carlo sampling of walks in H² that either reproduces or fails to reproduce the essential singularity in the correlation length predicted by the adapted RG flow.

Figures

Figures reproduced from arXiv: 2511.18510 by Daniil Fedotov, Sergei Nechaev.

Figure 1
Figure 1. Figure 1: Flowchart: Three faces of random walks in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Brownian bridge above the disc of radius [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: Expectation ∆rE(R) as a function of R for stretched paths of length L ≡ t = cR in double-logarithmic coordinates; Right: Comparison of the distribution Ω(r) with const Ai2 (βR−1/3 r + ˜a1) for R = 100, c = 10 and mixed (third-type) boundary conditions with κ = 0.2, ˜a1 ≈ −1.953, β ≈ 0.357 10 8 × 10 1 9 × 10 1 2 R 10 1 8.5 × 10 0 9 × 10 0 9.5 × 10 0 1.05 × 10 1 r (R) Semicircle: log-log slope 0.37 (c=… view at source ↗
Figure 4
Figure 4. Figure 4: Left: Expectation ∆rE(R) as a function of R for stretched paths of length L ≡ t = cR in double-logarithmic coordinates; Right: Comparison of the distribution Ω(r) with const Ai2 (βR−1/3 r + a1) for R = 100, c = 10 and Dirichlet boundary conditions. β ≈ 0.371 In both cases, (i) and (ii), the distribution function and the average span of radial fluc￾tuations of stretched paths at = cR for a = 1 and c = 10 ha… view at source ↗
read the original abstract

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that continuous random walks (diffusion) in the Poincaré hyperbolic upper half-plane H² unify three phenomena: (i) the non-analytic divergence of the correlation length at the BKT transition, obtained by adapting Efimov RG equations to diffusion in H²; (ii) KPZ-type behavior in the survival probability of stretched random walks near the boundary, derived via scaling arguments, WKB approach, and numerical analysis; and (iii) Lifshitz tails in 1D rare-event statistics, obtained via an instanton approach in deterministic large-deviation statistics. A conjecture is made that BKT-like physics is dominated by trajectories in the large-deviation stretched regime.

Significance. If the RG adaptation is shown to be faithful, the work would provide a geometric unification of BKT, KPZ, and Lifshitz-tail phenomena through hyperbolic diffusion, with the multi-method approach (RG, scaling/WKB/numerics, instantons) offering cross-checks where derivations are explicit.

major comments (1)
  1. [Abstract (RG adaptation paragraph)] Abstract (RG adaptation paragraph): The central claim for the BKT divergence rests on directly adapting the Efimov RG equations to diffusion in H² without additional terms from the hyperbolic Laplacian or Poincaré metric altering the beta functions. Explicit verification that the mapping is term-by-term faithful (i.e., no geometry-induced corrections appear at the same order in the scaling regime) is required; without it, reproduction of the essential singularity remains unverified and the unification for (i) is not load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and the identification of a point requiring clarification in our treatment of the BKT divergence. We address the single major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: The central claim for the BKT divergence rests on directly adapting the Efimov RG equations to diffusion in H² without additional terms from the hyperbolic Laplacian or Poincaré metric altering the beta functions. Explicit verification that the mapping is term-by-term faithful (i.e., no geometry-induced corrections appear at the same order in the scaling regime) is required; without it, reproduction of the essential singularity remains unverified and the unification for (i) is not load-bearing.

    Authors: We agree that the manuscript states the adaptation of the Efimov RG equations but does not supply an explicit term-by-term derivation showing that the hyperbolic Laplacian and Poincaré metric introduce no corrections to the beta functions at the scaling order of interest. The adaptation is motivated by the conformal equivalence of the Poincaré metric to the flat metric together with the form of the diffusion operator, which we expect to preserve the relevant RG structure. Nevertheless, the referee is correct that this expectation must be verified explicitly. In the revised version we will add a dedicated appendix that starts from the diffusion equation on H², performs the change of variables to the Efimov variables, and demonstrates that geometry-induced terms either vanish or appear only at higher order in the scaling regime, thereby confirming reproduction of the essential singularity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external RG adaptation and independent scaling/WKB/numerical/instanton methods within the model.

full rationale

The paper adapts RG equations from the Efimov effect (external prior work) to reproduce BKT divergence in H² diffusion, then derives KPZ behavior via scaling arguments, WKB, and numerics, and Lifshitz tails via instanton approach. The final conjecture on trajectory dominance is explicitly labeled as such and does not serve as a load-bearing derivation step. No self-definitional loops, fitted inputs renamed as predictions, self-citation load-bearing premises, uniqueness theorems, or smuggled ansatzes appear in the provided text. The model is self-contained against the cited external benchmarks and internal calculations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; full text would be required to enumerate all free parameters and background assumptions.

axioms (1)
  • domain assumption Diffusion in the Poincaré upper half-plane obeys the same conformal properties used in the Efimov RG flow.
    Invoked when the authors adapt the RG equations to H².

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