Random Monomial Ideals

TL;DR

This paper introduces probabilistic models for random monomial ideals inspired by random graphs, analyzing their properties and thresholds, and proposing conjectures about their algebraic invariants.

Contribution

It develops new probabilistic models for monomial ideals and provides theoretical results and conjectures on their algebraic properties.

Findings

Derived probability distributions and expectations for monomial ideals.

Identified thresholds for properties like Cohen-Macaulayness and regularity.

Proposed conjectures supported by experimental evidence.

Abstract

Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers, and present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen-Macaulayness of random monomial ideals.