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arxiv: 2506.14694 · v2 · submitted 2025-06-17 · 🧮 math.CO · math.AT· math.PR

The homology torsion growth of determinantal hypertrees

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

Pith reviewed 2026-05-19 08:56 UTC · model grok-4.3

classification 🧮 math.CO math.ATmath.PR
keywords homology torsiondeterminantal hypertreesrandom hypertreestorsion growthconvergence in probabilityhigh-dimensional complexescombinatorial topologyhomology of hypertrees
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The pith

For a random d-dimensional determinantal hypertree on n vertices, the log of the torsion in its (d-1)th homology group divided by binom(n,d) converges in probability to a constant c_d bounded between (1/2)log((d+1)/e) and (1/2)log(d+1).

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

The paper proves that this normalized log torsion converges in probability to some constant c_d for any fixed dimension d at least 2. The constant satisfies explicit lower and upper bounds that depend only on d. A reader would care because the result gives a precise asymptotic description of how large the torsion typically becomes in these random high-dimensional structures as the number of vertices grows. The convergence means that for most such hypertrees the torsion is exponentially large in the number of possible d-subsets, with a growth rate that is known up to a factor of roughly 2 in the log scale.

Core claim

We prove that log|H_{d-1}(T_n, Z)| / binom(n,d) converges in probability to a constant c_d, which satisfies (1/2) log((d+1)/e) ≤ c_d ≤ (1/2) log(d+1).

What carries the argument

The probability measure on d-dimensional determinantal hypertrees T_n, which induces a distribution on their homology groups whose torsion can be normalized and shown to concentrate.

If this is right

  • The typical size of the torsion is exp(c_d * binom(n,d) + o(binom(n,d))), so it grows exponentially with the number of d-subsets.
  • The growth rate c_d is sandwiched between the two explicit logarithmic expressions for every fixed d >= 2.
  • The same limit holds with high probability under the determinantal measure, not just on average.
  • The result applies uniformly across all dimensions d once n is large enough relative to d.

Where Pith is reading between the lines

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

  • The bounds suggest that c_d is roughly (1/2)log(d) for large d, which might be compared with growth rates in other random simplicial complexes.
  • Exact computation of c_d itself would require finer analysis of the underlying determinantal probabilities beyond the sandwich bounds.
  • The convergence in probability could be strengthened to almost-sure convergence or to a central limit theorem for the fluctuations.

Load-bearing premise

The random model of determinantal hypertrees is well-defined so that the homology torsion is a finite positive integer for almost every realization.

What would settle it

Numerical computation of the average of log |H_{d-1}(T_n, Z)| / binom(n,d) for d=2 and n up to several hundred, checking whether the values stabilize inside the interval from (1/2)log(3/e) to (1/2)log(3).

read the original abstract

Fix a dimension $d\ge 2$, and let $T_n$ be a random $d$-dimensional determinantal hypertree on $n$ vertices. We prove that \[\frac{\log|H_{d-1}(T_n,\mathbb{Z})|}{{{n\choose {d}}}}\] converges in probability to a constant $c_d$, which satisfies \[\frac{1}2 \log\left(\frac{d+1}e\right)\le c_d\le \frac{1}2 \log\left(d+1\right) .\]

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 / 1 minor

Summary. The manuscript proves that for a random d-dimensional determinantal hypertree T_n on n vertices, the normalized quantity log |H_{d-1}(T_n, Z)| / binom(n,d) converges in probability to a constant c_d satisfying the explicit bounds (1/2) log((d+1)/e) ≤ c_d ≤ (1/2) log(d+1).

Significance. If the result holds, it supplies a law-of-large-numbers statement for homology torsion growth under the determinantal model, which is notable for permitting exact combinatorial calculations. The non-vacuous matching bounds on c_d are a clear strength, as they arise directly from the model without fitted parameters and give concrete control on the growth rate.

major comments (1)
  1. [Model definition and main theorem statement] The central convergence statement presupposes that the determinantal construction defines a probability measure supported precisely on d-dimensional hypertrees (vanishing lower homology) for which H_{d-1}(T_n, Z) is finite torsion almost surely. This well-definedness and normalization must be established explicitly (see the model definition and the statement of the main theorem); otherwise log |H_{d-1}| is not a well-defined random variable and the normalized convergence claim does not make sense.
minor comments (1)
  1. [Abstract] The binomial coefficient in the abstract is typeset as {{n choose d}}; standard LaTeX binom notation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to strengthen the explicit justification of the model.

read point-by-point responses

  1. Referee: The central convergence statement presupposes that the determinantal construction defines a probability measure supported precisely on d-dimensional hypertrees (vanishing lower homology) for which H_{d-1}(T_n, Z) is finite torsion almost surely. This well-definedness and normalization must be established explicitly (see the model definition and the statement of the main theorem); otherwise log |H_{d-1}| is not a well-defined random variable and the normalized convergence claim does not make sense.

    Authors: We agree that an explicit statement of well-definedness is needed for full rigor. The determinantal construction is defined to be supported on d-dimensional hypertrees (hence with vanishing lower-dimensional homology) by the properties of the underlying determinantal process on the simplicial matroid; finiteness of H_{d-1} torsion follows because every such complex is finite. Nevertheless, to meet the referee's request, we will add a short proposition immediately after the model definition that proves the measure is supported precisely on hypertrees with finite H_{d-1} torsion almost surely, and we will update the main theorem statement to reference this fact. These changes will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Convergence result derived from model definition with no reduction to inputs by construction

full rationale

The paper defines the determinantal hypertree model and proves a law of large numbers for the normalized log torsion order via direct analysis of the random complex. No equation equates the target limit to a fitted parameter, no self-citation supplies a uniqueness theorem that forces the result, and the stated bounds on c_d are obtained from separate volume or entropy estimates rather than tautological renaming. The derivation chain is self-contained against the model's probability measure and standard homological algebra; the a.s. finiteness of torsion is part of the model's well-definedness rather than an output assumed in the proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard definition of determinantal hypertrees and the finiteness of torsion in the relevant homology groups; no free parameters or invented entities are introduced in the abstract statement.

axioms (2)
  • domain assumption The random determinantal hypertree model is a well-defined probability space on which homology is defined and torsion is finite almost surely.
    Required for |H_{d-1}(T_n,Z)| to be a positive integer whose normalized log can converge.
  • standard math Standard properties of simplicial homology with integer coefficients hold for the hypertrees under consideration.
    Background fact from algebraic topology invoked to make sense of H_{d-1}(T_n,Z).

pith-pipeline@v0.9.0 · 5613 in / 1457 out tokens · 37123 ms · 2026-05-19T08:56:02.538244+00:00 · methodology

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Reference graph

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