Asymptotic properties of random monomial ideals

2026-05-05

AI Summary

This study investigates the asymptotic density phase transition behavior of LCM lattices associated with random monomial ideals. Combining combinatorial algebra, probabilistic methods, and large-scale computational experiments, the work reveals a sharp threshold phenomenon—rather than a smooth transition—in LCM lattice density, identifying three distinct regimes: low-density, high-density, and a narrow transition window. The primary contributions include uncovering a strong negative correlation between the number of generators and LCM lattice density, and proving that increasing the degree of generators significantly lowers the critical threshold at which density abruptly drops. Building on these findings, the authors formulate a conjecture regarding the density phase transition for squarefree random monomial ideals, offering a novel perspective on their structural complexity and the emergence mechanism of giant components analogous to those in hypergraph theory.

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.

Problem

• random monomial ideals
• LCM-lattice density
• asymptotic properties
• phase transition
• redundancy

Innovation

• LCM-lattice
• random monomial ideals
• phase transition
• poset density
• asymptotic statistics