Asymptotic properties of random monomial ideals
For algebraic geometers and combinatorialists, this work provides a statistical perspective on monomial ideals, showing that redundancy and complexity indicators concentrate into distinct typical regimes, analogous to phase transitions in hypergraphs.
This paper studies asymptotic properties of random monomial ideals, finding that the LCM-lattice density exhibits a sharp phase transition between low-density Taylor-like and high-density redundant regimes, with a narrow transition window. Increasing generator degree lowers the probability threshold for this density drop.
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.