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arxiv: 2605.22103 · v1 · pith:4CYWCERMnew · submitted 2026-05-21 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Berezinskii-Kosterlitz-Thouless-type Transition in Site Percolation on the Diamond Hierarchical Lattice

Takehisa Hasegawa , Kazuki Wataya , Tomoaki Nogawa This is my paper

Pith reviewed 2026-05-22 04:28 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords site percolationdiamond hierarchical latticeBerezinskii-Kosterlitz-Thouless transitionessential singularitycritical phaserenormalization groupfractal networkcorrelation length
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The pith

Site percolation on the diamond hierarchical lattice shows a Berezinskii-Kosterlitz-Thouless-type transition with an essential singularity in the correlation length.

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

The paper demonstrates that site percolation on this finite-dimensional fractal lattice lacks a standard percolating phase transition. Instead, below a critical occupation probability p_c the system is non-percolating, while above it enters a critical phase where the largest cluster grows subextensively as a power of system size with a continuously varying exponent. Analysis of the renormalization-group recursion reveals that the correlation length diverges with a BKT-type essential singularity exp(const / sqrt(p_c - p)) as p approaches p_c from below, supported by finite-size scaling collapse. This indicates that critical phases can arise in percolation on finite-dimensional networks due to the persistent relevance of site dilution.

Core claim

The central claim is that site percolation on the diamond hierarchical lattice exhibits a nonpercolating phase for p < p_c and a critical phase for p > p_c, in which the largest cluster size scales subextensively as N^{ψ(p)} with ψ(p) varying continuously with p. The correlation length displays a Berezinskii-Kosterlitz-Thouless-type essential singularity ξ(p) ∼ exp(const / √(p_c − p)) for p → p_c^−, derived from the renormalization-group recursion relation and confirmed through finite-size scaling analyses.

What carries the argument

The renormalization-group recursion relation obtained from the exact generating-function analysis, which governs the flow of the percolation probability and reveals the essential singularity near p_c.

If this is right

  • The largest cluster remains subextensive in the critical phase, scaling with a p-dependent fractal exponent.
  • Critical phases in percolation can occur on finite-dimensional networks without requiring exponential volume growth.
  • The transition is driven by site dilution remaining relevant under renormalization-group transformations.
  • Finite-size scaling shows excellent data collapse consistent with the essential singularity.

Where Pith is reading between the lines

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

  • This behavior may generalize to other hierarchical lattices where site and bond percolation differ in their fixed-point structure.
  • Similar essential singularities could be searched for in percolation on other fractals or disordered media to test the role of dilution relevance.
  • Extensions to dynamics or other critical exponents might reveal more about the nature of this critical phase.

Load-bearing premise

The derived renormalization-group recursion relation accurately captures the leading behavior near p_c without being altered by higher-order corrections or lattice-specific effects.

What would settle it

Numerical computation of the correlation length on successively larger diamond hierarchical lattices for p slightly below p_c, checking whether it follows the predicted exp(const/sqrt(delta p)) form or deviates to a power-law divergence.

Figures

Figures reproduced from arXiv: 2605.22103 by Kazuki Wataya, Takehisa Hasegawa, Tomoaki Nogawa.

Figure 1
Figure 1. Figure 1: FIG. 1. Construction of the DHL. (a) The graph [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Conditional contact probability [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Normalized mean root-cluster size ¯s [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Log-log plot of the correlation function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Susceptibility [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Finite-size scaling plots of (a) the conditional contact probability [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Cumulative cluster-size distribution [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
read the original abstract

We study site percolation on the diamond hierarchical lattice, a finite-dimensional fractal network, using an exact generating-function analysis. In contrast to bond percolation, site percolation on this lattice does not undergo a transition from a nonpercolating phase to a percolating phase. Instead, the system exhibits a nonpercolating phase for $p<p_{\rm c}$ and a critical phase for $p>p_{\rm c}$. In the critical phase, the size of the largest cluster remains subextensive, scaling as $N^{\psi(p)}$, where the fractal exponent $\psi(p)$ varies continuously with $p$. By analyzing the renormalization-group recursion relation in the vicinity of $p_{\rm c}$, we show that the correlation length exhibits a Berezinskii-Kosterlitz-Thouless-type essential singularity, $\xi(p)\sim \exp \left({\rm const}/\sqrt{p_{\rm c}-p}\right)$ for $p \to p_{\rm c}^-$, which is further confirmed by finite-size scaling analyses showing excellent data collapse. These results demonstrate that critical phases in percolation can emerge even on finite-dimensional networks and that exponential volume growth is not necessary for such phases to appear. We argue that the critical phase on the diamond hierarchical lattice stems from site dilution remaining relevant under renormalization.

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

2 major / 2 minor

Summary. The paper analyzes site percolation on the diamond hierarchical lattice via exact generating functions and renormalization-group recursions. It reports that, unlike bond percolation, there is no percolation transition; instead a non-percolating phase exists for p < p_c and a critical phase for p > p_c in which the largest cluster is sub-extensive (N^ψ(p) with continuously varying ψ(p)). Near p_c the correlation length is claimed to exhibit a BKT-type essential singularity ξ(p) ∼ exp(const/√(p_c−p)) for p → p_c^−, supported by finite-size scaling and data collapse. The authors attribute the critical phase to site dilution remaining relevant under renormalization.

Significance. If the RG analysis is confirmed, the result shows that BKT-like essential singularities can appear in percolation on finite-dimensional hierarchical lattices without requiring exponential volume growth, and that critical phases with continuously varying exponents are possible in site percolation. The exact generating-function derivation and the explicit recursion provide a clean, parameter-free route to the singularity; the reported data collapse adds numerical support. This would be a notable addition to the literature on percolation on fractals and on the conditions for BKT-type transitions in disordered systems.

major comments (2)
  1. [RG recursion analysis near p_c] § on RG flow near p_c (the paragraph containing the expansion of p′=f(p)): the derivation of the BKT singularity assumes that the leading nonlinearity after the fixed point p_c (where f′(p_c)=1) is quadratic with the correct sign and that higher-order (cubic and beyond) terms generated by the site-percolation generating functions are irrelevant. An explicit Taylor expansion of the recursion up to at least cubic order, together with the numerical values of the coefficients, is needed to confirm that the quadratic term indeed dominates the flow and produces the claimed essential singularity.
  2. [Finite-size scaling analyses] Finite-size scaling section (the data-collapse figures and the associated scaling ansatz): the collapse is performed assuming the BKT form ξ ∼ exp(c/√(p_c−p)). While visually excellent, the collapse alone does not independently establish the asymptotic singularity; a direct numerical extraction of the correlation length from the generating functions for successively larger generations, followed by a fit to the essential-singularity form, would strengthen the claim.
minor comments (2)
  1. [Methods] The definition of the correlation length ξ in terms of the generating functions should be stated explicitly (e.g., via the second moment or the cluster-size distribution) so that the connection to the RG flow is unambiguous.
  2. [Introduction] Notation: the fractal exponent ψ(p) is introduced without an equation number; adding a numbered definition would improve readability when it is later compared with the RG predictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments point by point below.

read point-by-point responses

  1. Referee: [RG recursion analysis near p_c] § on RG flow near p_c (the paragraph containing the expansion of p′=f(p)): the derivation of the BKT singularity assumes that the leading nonlinearity after the fixed point p_c (where f′(p_c)=1) is quadratic with the correct sign and that higher-order (cubic and beyond) terms generated by the site-percolation generating functions are irrelevant. An explicit Taylor expansion of the recursion up to at least cubic order, together with the numerical values of the coefficients, is needed to confirm that the quadratic term indeed dominates the flow and produces the claimed essential singularity.

    Authors: We agree with the referee that an explicit Taylor expansion up to cubic order, including numerical coefficients, is important to confirm the dominance of the quadratic nonlinearity. We have performed this expansion based on the site-percolation generating functions and included the results, along with the coefficient values, in the revised manuscript. This addition verifies that the quadratic term dominates and supports the BKT-type singularity. revision: yes

  2. Referee: [Finite-size scaling analyses] Finite-size scaling section (the data-collapse figures and the associated scaling ansatz): the collapse is performed assuming the BKT form ξ ∼ exp(c/√(p_c−p)). While visually excellent, the collapse alone does not independently establish the asymptotic singularity; a direct numerical extraction of the correlation length from the generating functions for successively larger generations, followed by a fit to the essential-singularity form, would strengthen the claim.

    Authors: We thank the referee for this recommendation. To provide independent confirmation of the asymptotic form, we have extracted the correlation length directly from the generating functions for successively larger generations and fitted it to the essential-singularity expression. The results of this analysis have been added to the revised finite-size scaling section, complementing the data-collapse analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: singularity emerges from exact RG flow on hierarchical lattice

full rationale

The recursion relation is obtained directly by composing the site-percolation generating functions on the diamond hierarchical lattice geometry; the fixed-point condition f'(p_c)=1 and the leading quadratic nonlinearity are computed from those same functions. Integration of the resulting differential RG equation then produces the essential singularity without any fitted parameters, ansatz, or self-citation that carries the central claim. Finite-size scaling collapse is presented as numerical corroboration rather than as the source of the functional form. The derivation is therefore self-contained against the lattice definition and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central results rest on the exact solvability of the hierarchical lattice via generating functions and the assumption that site dilution remains relevant under repeated renormalization steps; no new particles or forces are introduced.

free parameters (1)
  • p_c
    The percolation threshold p_c is located as the unstable fixed point of the renormalization-group recursion relation derived from the generating functions.
axioms (1)
  • domain assumption The diamond hierarchical lattice admits an exact generating-function representation that closes under renormalization.
    Invoked to enable the recursion analysis described in the abstract.

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Reference graph

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