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arxiv: 2606.28890 · v1 · pith:WYAZJWEHnew · submitted 2026-06-27 · 🧮 math.CO · math.PR

Logarithmic convergence of finite projective planes

M\'arton Borb\'enyi , Panna T\'imea Fekete , Aranka Hru\v{s}kov\'a , Ander Lamaison This is my paper

Pith reviewed 2026-06-30 09:31 UTC · model grok-4.3

classification 🧮 math.CO math.PR
keywords log-convergenceprojective planesincidence graphsfinite fieldsrandom graphsgraph sequencesSzegedy convergence
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The pith

Incidence graphs of projective planes over finite fields log-converge to the same limit as a specific random graph model.

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

The paper investigates log-convergence for sequences of graphs, a concept introduced by Balázs Szegedy. It establishes that the incidence graphs arising from projective planes defined over finite fields form a sequence that satisfies this form of convergence. The limit object obtained matches the limit produced by one particular random graph model. A reader would care because this provides an explicit, deterministic family of graphs whose large-scale behavior aligns with a random construction in this convergence sense, answering an open question posed by Szegedy.

Core claim

The sequence of incidence graphs of projective planes over finite fields log-converges, and the limit coincides with that of a particular random graph model.

What carries the argument

Log-convergence of graph sequences, applied to the incidence graphs of projective planes over finite fields.

If this is right

  • Szegedy's Question 4 receives an affirmative answer.
  • The incidence graphs of these planes achieve the same limiting object as the chosen random model.
  • Deterministic constructions from finite geometry can realize the limiting behavior previously associated only with random models.
  • The log-convergence framework treats these geometric graphs and the random model as equivalent at the limit level.

Where Pith is reading between the lines

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

  • Other families of algebraic graphs, such as those from higher-dimensional geometries, might be checked for the same log-convergence property.
  • Explicit small-field examples could be used to numerically approximate the rate at which the sequence approaches the limit.
  • The matching limit suggests the random model may capture some averaged or asymptotic property shared by all such planes.

Load-bearing premise

The log-convergence definition applies directly to these incidence graphs without needing extra regularity conditions.

What would settle it

A computation or limit calculation in which the projective-plane incidence graphs and the random model produce different values under the log-convergence measure.

Figures

Figures reproduced from arXiv: 2606.28890 by Ander Lamaison, Aranka Hru\v{s}kov\'a, M\'arton Borb\'enyi, Panna T\'imea Fekete.

Figure 1
Figure 1. Figure 1: Depending on the values of the parameters [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Point-line arrangements corresponding to Pappus’s theorem (left) and Desargues’s theorem [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

In this paper, we study the so-called log-convergence of graphs defined by Bal\'azs Szegedy (arXiv:1504.00858). We answer his Question 4 affirmatively: the sequence of incidence graphs of projective planes over finite fields log-converges, and the limit coincides with that of a particular random graph model.

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 log-convergence of graphs in the sense of Szegedy (arXiv:1504.00858) and affirmatively answers Question 4: the sequence of incidence graphs of projective planes over finite fields log-converges, and the limit object coincides with that arising from a specified random graph model.

Significance. If the central claim holds, the result supplies a concrete affirmative resolution to an open question in the Szegedy framework, furnishing a deterministic sequence from finite geometry whose log-limit matches a random-graph limit. This supplies an explicit, non-random example in the theory and may facilitate further comparisons between algebraic/combinatorial constructions and probabilistic models under log-convergence.

minor comments (2)
  1. The abstract states the main result but supplies no proof outline, key definitions, or verification steps; a one-paragraph sketch of the argument (e.g., how the Szegedy definition is applied to the incidence graphs and how the limit is identified) would improve readability without altering the technical content.
  2. Notation for the random graph model whose limit is claimed to coincide should be introduced explicitly in the introduction or §2, together with a precise reference to the model in Szegedy's paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report provides no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper invokes Szegedy's external definition of log-convergence (arXiv:1504.00858, different author) to establish that incidence graphs of finite-field projective planes converge to the same limit object as a specified random-graph model, thereby answering an open question. No self-citations appear in the load-bearing steps, no parameter is fitted and then relabeled as a prediction, and no uniqueness theorem or ansatz is imported from the authors' own prior work. The central claim therefore rests on independent application of the cited framework rather than on any definitional reduction or self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is provided in the abstract.

pith-pipeline@v0.9.1-grok · 5590 in / 995 out tokens · 44127 ms · 2026-06-30T09:31:45.535177+00:00 · methodology

discussion (0)

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Reference graph

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