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arxiv: 2606.17895 · v1 · pith:SKKH73FVnew · submitted 2026-06-16 · 🧮 math.CO · math.PR

Persistence diagrams of random triangular matrices over finite fields

Andr\'as M\'esz\'aros This is my paper

Pith reviewed 2026-06-27 00:14 UTC · model grok-4.3

classification 🧮 math.CO math.PR
keywords persistence diagramsrandom matricesfinite fieldsstochastic topologypersistent Betti numberslaw of large numbers
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The pith

An explicit formula gives the full distribution of the persistence diagram for row spans in a random lower triangular matrix over a finite field.

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

The paper considers an infinite lower triangular matrix whose on- and below-diagonal entries are drawn independently and uniformly from a fixed finite field. It tracks the nested sequence of subspaces formed by the spans of the first n rows and encodes the births and deaths of basis elements via the verbose persistence diagram. An explicit probability formula is derived for the entire diagram. A law of large numbers is proved for the lifetimes of these features, together with a description of the fluctuations of the persistent Betti numbers. A reader cares because the construction supplies an exactly solvable model inside stochastic topology where the algebraic randomness produces computable topological statistics.

Core claim

The verbose persistence diagram of the filtration given by the successive row spans admits an explicit distribution formula that depends only on the field size and the matrix dimensions; the lifetimes of the diagram satisfy a law of large numbers, and the persistent Betti numbers have fluctuations that can be described explicitly.

What carries the argument

The verbose persistence diagram, which records the birth and death times of each new basis vector contributed by successive rows.

If this is right

  • The expected number of features with any given lifetime can be computed in closed form.
  • The persistent Betti numbers converge almost surely after suitable centering and scaling.
  • The model yields exact formulas for the probability that a given dimension appears in the diagram at a given scale.

Where Pith is reading between the lines

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

  • The same algebraic mechanism may produce exact distributions for persistence in other filtrations built from random linear maps over finite fields.
  • Numerical checks on moderate-sized matrices would immediately confirm or refute the explicit formula.
  • The law of large numbers suggests that typical realizations become deterministic in the large-n limit after rescaling lifetimes by log n.

Load-bearing premise

The entries on and below the diagonal are independent and uniformly random elements of a fixed finite field.

What would settle it

Generate many independent finite truncations of the matrix, compute their persistence diagrams by row reduction, and test whether the empirical distribution of diagrams converges to the claimed explicit formula.

Figures

Figures reproduced from arXiv: 2606.17895 by Andr\'as M\'esz\'aros.

Figure 1
Figure 1. Figure 1: A sample of the verbose persistence diagram with the choice [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The persistence diagram above displays the birth and death times [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two sample input-output pairs for the procedure [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A random sample of the matrix A with the choice of q = 2. The blue cells correspond to elements of the verbose persistence diagram P. The orientation of the picture is chosen such that it is in alignment with the persistence diagram, that is, the bottom left cell displays A(1, 1), the cell above that displays A(1, 2). 3.3 A multi phase sampling algorithm The following lemma is clear from the description of… view at source ↗
Figure 5
Figure 5. Figure 5: The regions of the matrix A filled by the different phases of MultiPhaseSamplingA. • If b ′ j ̸= −∞, then Mr,j = Mr,j−1. • If b ′ j = −∞, then P(Mr,j = k | Mr,0, Mr,1, . . . , Mr,j−1) =    q −Mr,j−1 if k = Mr,j−1, 1 − q −Mr,j−1 if k = Mr,j−1 − 1, 0 otherwise. In agreement with our earlier definitions, let T ′ 0 = 0, T ′ i = min{n > T′ i−1 : b ′ n = −∞} for i > 0. Then our previous observations on the … view at source ↗
read the original abstract

Let us consider a random infinite lower triangular matrix, where the entries on and below the diagonal are i.i.d. uniform random elements of a fixed finite field. We investigate the evolution of the span of the first $n$ rows of this matrix as $n$ grows. Many properties of this evolving subspace can be captured with the help of the verbose persistence diagram, which is a standard tool in stochastic topology and topological data analysis. We give an explicit formula for the distribution of the persistence diagram. We prove a law of large numbers for the distribution of lifetimes. We also describe the fluctuations of the persistent Betti numbers.

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 / 3 minor

Summary. The paper considers an infinite lower-triangular random matrix over a fixed finite field whose entries on and below the diagonal are i.i.d. uniform. It tracks the nested sequence of row spans and encodes their evolution via the verbose persistence diagram. The central claims are an explicit formula for the distribution of this diagram, a law of large numbers for the lifetimes, and fluctuation results for the persistent Betti numbers.

Significance. An explicit distributional formula for the persistence diagram in this discrete random-subspace model would be a notable advance; most results in stochastic topology supply only asymptotics or moments. The finite-field setting permits exact combinatorial counting, which, if carried through, strengthens the link between persistent homology and algebraic combinatorics. The LLN and fluctuation statements would furnish a complete first-order and second-order picture for the lifetimes and Betti numbers.

minor comments (3)
  1. [Abstract] The abstract introduces the 'verbose persistence diagram' without a one-sentence reminder of its definition or a pointer to the section where it is formalized; a brief parenthetical would improve accessibility.
  2. Notation for the finite field and its cardinality is introduced only in the model statement; repeating the symbol (e.g., 𝔽_q) in the statements of the main theorems would make the dependence on q transparent.
  3. The law-of-large-numbers statement for lifetimes is phrased in terms of 'distribution of lifetimes'; a short clarification whether this is the empirical measure on the diagram or the marginal on each lifetime would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly summarizes the main contributions: the explicit distribution of the verbose persistence diagram, the LLN for lifetimes, and the fluctuation results for persistent Betti numbers.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a standard i.i.d. uniform model on lower-triangular entries over a finite field and states that it derives an explicit formula for the distribution of the verbose persistence diagram, an LLN for lifetimes, and fluctuation results for persistent Betti numbers. No equations, self-citations, or fitted parameters are visible in the provided abstract or reader summary that would reduce any claimed prediction back to the inputs by construction. The derivation is presented as a direct probabilistic computation on the evolving row spans, which is self-contained against the model assumptions without load-bearing self-references or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; ledger entries are therefore minimal and provisional.

axioms (1)
  • domain assumption Entries on and below the diagonal are i.i.d. uniform over a fixed finite field.
    Stated in the first sentence of the abstract as the generative model.

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Reference graph

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