Asymptotic properties of random monomial ideals

TL;DR

This paper investigates the asymptotic behavior of random monomial ideals, revealing sharp threshold phenomena and phase transitions in their LCM-lattice density across different regimes.

Contribution

It introduces a statistical perspective on the redundancy and density of LCM-lattices in random monomial ideals, highlighting phase transition behavior.

Findings

LCM-lattice density exhibits sharp threshold behavior.

Negative correlation between number of generators and density.

Density drops occur at lower thresholds with higher generator degrees.

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.