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arxiv: 2408.10927 · v6 · submitted 2024-08-20 · 🧮 math.PR

Critical percolation on slabs with random columnar disorder

Matheus B. Castro , R\'emy Sanchis , Roger W.C. Silva This is my paper

Pith reviewed 2026-05-23 21:59 UTC · model grok-4.3

classification 🧮 math.PR
keywords bond percolationslabsrenewal processcolumnar disorderpower-law tailscritical probabilityinhomogeneous percolation
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The pith

If the random columns are selected with sufficiently heavy power-law tails, percolation occurs for p below p_c whenever q exceeds p_c on finite-height slabs.

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

This paper examines a percolation model on positive half-space slabs of height k where certain vertical columns are chosen randomly via a renewal process with power-law interarrival tails of exponent φ greater than 1. On these selected columns the vertical edges open with probability q while all other edges open with probability p. The central result establishes that for all φ large enough depending only on k, percolation takes place as soon as q is larger than the critical probability of the homogeneous slab, even when p is chosen smaller than that same critical value. A sympathetic reader would care because this demonstrates that sparse but strong inhomogeneities can induce percolation in a regime where the uniform model does not percolate.

Core claim

For all sufficiently large φ depending solely on k, if q > p_c(S^+_k) then there exists p < p_c(S^+_k) such that the model percolates.

What carries the argument

The renewal process whose inter-arrival times have power-law tails with exponent φ that determines which columns receive the modified opening probability q for their vertical bonds.

If this is right

  • The phase transition in the (p, q) plane occurs at p values strictly below p_c when q > p_c for large φ.
  • Enhancing only a sparse random set of columns is sufficient to achieve percolation below the homogeneous threshold.
  • The result depends on the tail exponent φ being large enough relative to the slab height k.

Where Pith is reading between the lines

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

  • Similar effects might occur in other models with long-range correlations induced by heavy-tailed renewals.
  • The dependence of the required φ on k suggests that thicker slabs need even heavier tails to see the effect.

Load-bearing premise

The power-law exponent φ of the column selection renewal process must exceed some threshold that depends only on the slab height k.

What would settle it

A counterexample where for some k and sufficiently large φ, percolation fails to occur for any p < p_c when q > p_c would disprove the assertion.

Figures

Figures reproduced from arXiv: 2408.10927 by Matheus B. Castro, R\'emy Sanchis, Roger W.C. Silva.

Figure 1
Figure 1. Figure 1: Sketch of the sets Bn,LBn,RBn,LSn,RSn,BSn and T Sn. Define also the rectangles HRn = [−n,3n]× [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two-dimensional sketch of Dn(0) (continuous line), Hn(0) (dotted dark line) and Vn(0) (dashed light line). Let Jn = {i ∈ Z+ : I i n is λ-good}, Un = S i∈Jn I i n, and write Un = Un ×Z. We shall associate to every edge hx,yi ∈ E(Z2 +) the set Bn(x) ∪ Bn(y) ⊆ S + k . We say hx,yi ∈ E(Z2 +) is λ-favored if Bn(x) ∪ Bn(y) ⊆ Un. Otherwise, we say hx,yi is λ-unfavored. Hereafter, when there is no possibility of c… view at source ↗
Figure 3
Figure 3. Figure 3: A sketch of the sets Bn and C(f). Proposition 2. For each ρ < 1, there exists λ(ρ) > 0 such that the following holds: for every ε > 0, there exists δn = δ(n,ε,λ) > 0 such that, for n large enough, P Λ pc−δn,pc+ε (An(f)) ≥ ρ, if f is λ-favored. (10) Remark 1. To prove Theorem 1, we will take ρ0 as in (42), λ = λ(ρ0,k) and let φ0(k) > 1/λ. The multiscale scheme we describe below requires that favored edges a… view at source ↗
Figure 4
Figure 4. Figure 4: The renormalized percolation and the original equ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The boxes BN , R′ N , R′′ N and QN . bound for the probability that a given edge is pivotal for the event V (BN ). In what follows, let Pp denote the law of a Bernoulli percolation process with parameter p. Proposition 4. Let τ > 0 and p > pc(S + k ). There are constants c4(τ ) > 0 and c5(τ ) > 0 such that for all N ≤ Lτ (p), X e∈E(R′′ N ) Pp({e piv. to V (BN )}) ≥ c4(τ )  p 2(1−p) 4+k  N 4 c5(τ) . Pro… view at source ↗
Figure 6
Figure 6. Figure 6: A sketch of the paths γa,γb and γc. Let π : Z2 × [0,k] → Z2 , π(x,y,z) = (x,y), be the projection of the slab onto Z2 . If γ is a path, let π(γ) be the projection of the vertices of γ onto Z2 . Observe that the path γ ∗ = γa ⊕ hx0,y0i ⊕ γb crosses the box BN vertically, and γc crosses it horizontally. Moreover, π(γa)∩ π(γc) , ∅. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The sets C1, R1 1 , R2 1 and their subsets. For i = 1,...,3, let Ci be the cone joining K1 i to a, for i = 4,...,8, let Ci be the cone joining K1 i to c, and for i = 9,10,11, let Ci be the cone joining K1 i to b. For all i = 1,..,11, let Ci = Ci ∩B′′ d and C∗ d = Ci ∩ ∂B′′ d . We refer the reader to [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: On the left, a two-dimensional sketch of the open pa [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two-dimensional sketch of the paths constructed i [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Chain of models showing edge-opening probabilit [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
read the original abstract

We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on $\mathbb{Z}_+\times\{0,\dots,k\}$. Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent $\phi>1$. Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability $q$ (respectively $1-q$), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability $p$ (respectively $1-p$). We prove that for all sufficiently large $\phi$ (depending solely on $k$), the following assertion holds: if $q>p_c(\mathbb{S}^+_k)$, then $p$ can be taken strictly smaller than $p_c(\mathbb{S}^+_k)$ in a manner that percolation still occurs.

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

0 major / 2 minor

Summary. The paper studies bond percolation on the half-slab S^+_k = Z_+ × Z_+ × {0,…,k} with random columnar disorder. Columns are selected by a renewal process whose inter-arrival times have power-law tails of exponent φ > 1. Selected columns have vertical edges open with probability q; all other edges are open with probability p. The central theorem asserts that for every fixed k and all φ sufficiently large (depending only on k), whenever q > p_c(S^+_k) there exists p < p_c(S^+_k) such that the disordered model percolates.

Significance. If the result holds, it supplies a rigorous example in which sufficiently heavy-tailed one-dimensional disorder lowers the percolation threshold below the homogeneous value p_c(S^+_k). The statement is parameter-free once φ is large enough and relies on standard tools of percolation and renewal theory; this combination of an explicit existence result with a clean dependence only on slab height k is a clear strength.

minor comments (2)
  1. The notation S^+_k is introduced in the abstract but the precise boundary conditions on the half-slab (especially the treatment of the Z_+ direction) should be restated explicitly in the introduction or §1 for readers unfamiliar with the model.
  2. The phrase 'percolation still occurs' in the abstract should be replaced by a precise statement (e.g., 'there is an infinite cluster with positive probability') to avoid any ambiguity about the percolation event.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its clean parameter dependence on slab height k and the use of standard tools. We appreciate the recommendation for minor revision and will incorporate any such changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pure existence theorem in percolation theory: it proves that for all φ sufficiently large (depending only on slab height k), the disordered model percolates at some p < p_c(S^+_k) whenever q > p_c(S^+_k). The argument is constructed from standard renewal-process tail estimates and slab percolation comparisons; no parameters are fitted to data, no quantity is defined in terms of the target conclusion, and no load-bearing step reduces to a self-citation or ansatz smuggled from prior work by the same authors. The explicit dependence on large φ is stated as part of the theorem rather than an unexamined hypothesis, so the derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard axioms of probability and graph theory plus the definition of the renewal process; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Kolmogorov axioms of probability
    Used to define the product measure on the edge configurations.
  • domain assumption Existence of critical probability p_c for bond percolation on the slab
    Invoked when comparing q and p to p_c(S^+_k).

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