arxiv: 2605.03464 · v1 · submitted 2026-05-05 · 💻 cs.SC · math.AC· math.CO

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Asymptotic properties of random monomial ideals

Fatemeh Mohammadi , Sonja Petrovi\'c , Eduardo S\'aenz-de-Cabez\'on

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Pith reviewed 2026-05-07 04:14 UTC · model grok-4.3

classification 💻 cs.SC math.ACmath.CO

keywords monomial idealsLCM-latticerandom idealsposet densityphase transitionsasymptotic behavioralgebraic statistics

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The pith

The LCM-lattice density of random monomial ideals exhibits sharp threshold behavior that separates three distinct regimes rather than changing smoothly.

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

The paper studies how the density of the LCM-lattice behaves in families of random monomial ideals and in structured network models. Experimental sampling shows that this density drops abruptly as the number of generators grows, creating a low-density regime that resembles Taylor resolutions, a high-density regime full of redundancies, and only a narrow window between them. Raising the degree of the generators moves the location of the drop to smaller probabilities. The authors conjecture that equigenerated squarefree ideals undergo a genuine phase transition analogous to the sudden appearance of giant components in random hypergraphs. This statistical picture suggests that classical combinatorial invariants of ideals acquire typical, concentrated values inside natural random families.

Core claim

In random families of monomial ideals the poset density of the LCM-lattice undergoes sharp threshold behavior that abruptly separates a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window; the transition point moves to lower probabilities when generator degree increases. For equigenerated squarefree monomial ideals the density is conjectured to exhibit a phase transition of the same kind that governs the emergence of giant components in random hypergraphs.

What carries the argument

The poset density of the LCM-lattice, defined as the proportion of comparable pairs among the least common multiples of the generators, which serves as a combinatorial measure of redundancy and resolution complexity.

If this is right

Redundancy and resolution-complexity indicators concentrate into distinct typical regimes inside natural random families of monomial ideals.
The LCM-lattice functions as a statistical invariant whose value can be predicted from the number and degree of generators once the regime is known.
For equigenerated squarefree ideals the transition in density is expected to be as sharp as the percolation threshold in random hypergraphs.
Increasing generator degree shifts the entire threshold curve to lower probabilities, tightening the window of typical behavior.
Classical ideal-by-ideal invariants therefore admit an asymptotic counterpart that concentrates on a small number of typical values.

Where Pith is reading between the lines

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

The observed regimes may guide probabilistic algorithms that simplify random ideals by first detecting which density phase they occupy.
Similar threshold phenomena are likely to appear in other algebraic invariants such as Betti tables or syzygy modules when the same random models are applied.
The hypergraph analogy suggests that tools from random combinatorial structures could be imported to prove the conjectured phase transition for squarefree ideals.
Systematic sampling at larger scales could test whether the three-regime picture persists when the number of variables grows with the number of generators.

Load-bearing premise

The chosen random algebraic models and network constructions produce LCM-lattice densities that are representative of the asymptotic behavior of monomial ideals in general.

What would settle it

A smooth, gradual change in LCM-lattice density with increasing number of generators, instead of abrupt jumps that create three separated regimes, would falsify the sharp-threshold claim.

Figures

Figures reproduced from arXiv: 2605.03464 by Eduardo S\'aenz-de-Cabez\'on, Fatemeh Mohammadi, Sonja Petrovi\'c.

**Figure 1.** Figure 1: Graph density, lcm density, Betti density and length view at source ↗

**Figure 2.** Figure 2: Density and graph density for all graphs in view at source ↗

**Figure 4.** Figure 4: Linear regression plot between collection density view at source ↗

**Figure 3.** Figure 3: shows the density of each collection of 𝑑-subsets together with the lcm-density of the corresponding ideal. The lcmdensities are plotted on a logarithmic scale, while the collection densities are plotted on a linear scale. The plot shows an almost perfect inverse relationship between these two quantities. This behavior is quantified in view at source ↗

**Figure 5.** Figure 5: Mean and standard deviation of lcm-density for equigenerated random monomial ideals in degree 𝑑 = 2 in 𝑛 = 2 variables (left) and 𝑛 = 5 variables (right). For each value of 𝑛, 𝑑, 𝑝, the sample contains 100 monomial ideals. If one interprets the squarefree condition as a sampling restriction on samples from the I (𝑛, 𝐷, 𝑝) model, the results here further support our previous observation: that the size of t… view at source ↗

**Figure 6.** Figure 6: Mean and its standard error for lcm-density for equigenerated random monomial ideals generated from the model I (𝑛, 𝑑, 𝑝). For each value of 𝑛, 𝑑, 𝑝, the sample contains 100 monomial ideals. The lcm-density is shown on a log scale. Given that generating large samples of monomial ideals on many variables is non-trivial, we showcase the behavior of the mean lcmdensities for small 𝑛 and varying degrees 𝑑 for… view at source ↗

read the original abstract

This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density behaves across random algebraic models and structured networks. Experimental data reveal that the LCM-lattice exhibits sharp threshold behavior rather than changing smoothly. We observe a strong negative correlation between the number of generators and LCM-lattice density, abruptly separating three distinct regimes: a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window. We show that increasing the generator degree causes this density drop to occur at lower probability thresholds. We conclude by conjecturing that for equigenerated squarefree ideals, the LCM-lattice density undergoes a sharp phase transition, analogous to the emergence of giant components in hypergraphs. This suggests that the classical, ideal-by-ideal role of the LCM-lattice as a combinatorial invariant also admits a statistical/asymptotic counterpart: in natural random families, redundancy and resolution-complexity indicators concentrate into distinct typical regimes separated by a narrow transition window.

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.

Desk Editor's Note private letter to a colleague

Experiments show LCM-lattice density in random monomial ideals drops sharply with more generators and separates into three regimes, but the asymptotic sharpness claim lacks scaling support.

read the letter

The paper's main observation is that in several random models for monomial ideals the density of the LCM-lattice falls abruptly once the number of generators passes a threshold, producing a low-density Taylor-like regime, a narrow transition band, and a high-density redundant regime. They also note that raising the generator degree moves the drop to lower probabilities, and they conjecture that the squarefree equigenerated case mirrors the giant-component transition in random hypergraphs. That is the concrete new content: a statistical description of when the LCM-lattice becomes sparse or dense, framed as an extension of earlier redundancy work. The experiments appear to have been run on both purely random algebraic models and on structured networks, which gives the results some breadth. The negative correlation between generator count and lattice density is reported consistently enough to be worth recording. The conjecture itself is stated plainly as an open statement rather than a derived result, which keeps the claims proportionate. The soft spot is the lack of finite-size scaling. The abstract and stress-test note both indicate that the transition is described as sharp on the basis of finite samples, yet there is no mention of how the window width changes when the number of variables or the degree is increased systematically. Without those plots it is hard to rule out that the apparent abruptness is a moderate-size artifact rather than an asymptotic phase transition. Reproducibility details such as sample sizes, exact model parameters, and any statistical tests are also not summarized in the abstract, so a referee would need to see the full experimental protocol. The work is aimed at people who already care about random algebraic structures and average-case complexity in computational algebra. It is not trying to prove new theorems about individual ideals; it is trying to map out typical behavior. That audience would find the regime separation and the hypergraph analogy useful to discuss, even if the conjecture stays open. I would send it to peer review. The experimental findings are specific enough to be checked, the modeling choices are reasonable, and the direction connects algebraic combinatorics to probabilistic combinatorics in a way that has not been done before for monomial ideals. A few rounds of revision on the scaling analysis and experimental reporting would make the paper solid.

Referee Report

2 major / 2 minor

Summary. The manuscript examines asymptotic properties of random monomial ideals by studying the poset density of the LCM-lattice across random algebraic models and structured networks. Experimental observations indicate that this density exhibits sharp threshold behavior, with a strong negative correlation to the number of generators that separates three regimes (low-density Taylor-like, high-density redundant, and narrow transition). Increasing generator degree shifts the density drop to lower probability thresholds. The paper concludes with a conjecture that equigenerated squarefree ideals undergo a sharp phase transition in LCM-lattice density, analogous to the emergence of giant components in hypergraphs, implying that redundancy and resolution complexity concentrate into distinct typical regimes.

Significance. If the reported thresholds and regime separations are asymptotically robust, the work would establish a statistical counterpart to the classical combinatorial role of the LCM-lattice, linking monomial ideal theory to random hypergraph phenomena and offering predictions for typical resolution complexity in random families. The conjecture supplies a concrete, falsifiable statement that could guide further analytic or computational study. The experimental exploration of multiple models is a positive step toward identifying universal behavior, though the absence of scaling analysis limits the strength of the asymptotic claims.

major comments (2)

[Experimental results and conjecture] Experimental section (results on threshold behavior): The claim that the LCM-lattice 'exhibits sharp threshold behavior rather than changing smoothly' and that the density drop 'abruptly separates three distinct regimes' is load-bearing for the central statistical claim, yet the manuscript provides no finite-size scaling analysis. No plots or tables show how the transition window width varies with the number of variables n or generator degree, nor any quantitative measure (e.g., window width scaling as O(1/n) or faster) to distinguish asymptotic phase transitions from finite-size sigmoidal effects. Without this, the observed negative correlation and regime separation could be artifacts of the sampled scales, weakening support for the conjecture on equigenerated squarefree ideals.
[Abstract and Section 4 (experimental setup)] Abstract and experimental description: The abstract states that 'experimental data reveal sharp thresholds and a strong negative correlation' but supplies no sample sizes, number of trials per parameter point, error bars, statistical tests for threshold detection, or explicit definitions of the random models (e.g., probability distributions over generators). These omissions make it impossible to evaluate reproducibility or the robustness of the reported regime separation, which is central to the paper's contribution.

minor comments (2)

[Introduction] Notation for LCM-lattice density is introduced without a clear equation reference in the early sections; adding an explicit definition (e.g., as the normalized height or width of the poset) would improve readability.
[Concluding conjecture] The analogy to giant components in hypergraphs is stated in the conjecture but lacks a brief reference to the relevant hypergraph literature (e.g., on threshold phenomena for k-uniform hypergraphs) to help readers connect the algebraic and combinatorial settings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which identify key areas where the experimental support for our claims can be strengthened. We address each major comment below and will incorporate revisions to improve the rigor and reproducibility of the manuscript.

read point-by-point responses

Referee: [Experimental results and conjecture] Experimental section (results on threshold behavior): The claim that the LCM-lattice 'exhibits sharp threshold behavior rather than changing smoothly' and that the density drop 'abruptly separates three distinct regimes' is load-bearing for the central statistical claim, yet the manuscript provides no finite-size scaling analysis. No plots or tables show how the transition window width varies with the number of variables n or generator degree, nor any quantitative measure (e.g., window width scaling as O(1/n) or faster) to distinguish asymptotic phase transitions from finite-size sigmoidal effects. Without this, the observed negative correlation and regime separation could be artifacts of the sampled scales, weakening support for the conjecture on equigenerated squarefree ideals.

Authors: We acknowledge that a quantitative finite-size scaling analysis is necessary to rigorously establish the asymptotic nature of the observed thresholds and to distinguish them from finite-size smoothing effects. Although our experiments were performed across a range of n and generator degrees, with the density drop visibly sharpening for larger n, the submitted manuscript does not include explicit scaling plots or fitted exponents for the transition window width. In the revised version we will add a new subsection to the experimental results that reports the window width (defined via the p-interval over which density falls from 0.8 to 0.2) as a function of n for several fixed degrees, together with a scaling fit. This additional analysis will provide direct quantitative support for the conjecture by showing that the transition sharpens in the large-n limit, consistent with the hypergraph giant-component analogy. revision: yes
Referee: [Abstract and Section 4 (experimental setup)] Abstract and experimental description: The abstract states that 'experimental data reveal sharp thresholds and a strong negative correlation' but supplies no sample sizes, number of trials per parameter point, error bars, statistical tests for threshold detection, or explicit definitions of the random models (e.g., probability distributions over generators). These omissions make it impossible to evaluate reproducibility or the robustness of the reported regime separation, which is central to the paper's contribution.

Authors: We agree that the experimental methodology must be documented with sufficient detail for reproducibility. While the random models are defined in Section 3, the abstract and experimental section do not explicitly list sample sizes, trial counts, error bars, or the precise statistical procedures used to locate thresholds. In the revision we will expand the abstract to note the experimental scale and will augment Section 4 with explicit statements of the probability distributions defining each random model, the number of independent trials performed per parameter point, error bars on all plots, and the method employed to detect thresholds (e.g., inflection-point fitting). These changes will enable readers to assess the robustness of the reported regime separations and correlations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are direct experimental observations and an explicit open conjecture

full rationale

The paper reports experimental sampling of random monomial ideals and structured networks, documenting a negative correlation between number of generators and LCM-lattice density that separates low-density, high-density, and transition regimes. These observations are presented as direct outputs of the chosen random models rather than any algebraic derivation or fitted parameter that is then relabeled as a prediction. The final statement is explicitly labeled a conjecture about phase-transition behavior in equigenerated squarefree ideals, kept separate from the data. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled in, and no step reduces the target density or threshold by construction to the input sampling procedure. The derivation chain is therefore self-contained as statistical exploration of specific families.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions from commutative algebra and on the modeling assumption that the chosen random families capture asymptotic behavior; no free parameters are fitted in the reported results, and no new entities are postulated.

axioms (2)

standard math Monomial ideals and their LCM-lattices obey the standard definitions and properties of commutative algebra.
Invoked throughout the description of the poset density and Taylor-like regimes.
domain assumption The random algebraic models and structured networks produce samples whose statistical properties reflect the typical asymptotic behavior of monomial ideals.
Required for the experimental observations of thresholds and correlations to support the conjecture about general families.

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Reference graph

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