Probabilities of random monomial ideals associated to large graphs

TL;DR

This paper introduces a probabilistic model for random monomial ideals derived from large graphs, analyzing their algebraic properties and asymptotic behaviors of invariants like dimension, regularity, and the v-number.

Contribution

It proposes a new graph-based probabilistic model for monomial ideals and derives asymptotic results for their algebraic invariants, including thresholds for normality and dimension.

Findings

Established conditions for normality in the model.

Derived threshold functions for Krull dimension.

Analyzed asymptotic behavior of regularity and v-number.

Abstract

Inspired by the Erd\H{o}s R\'enyi model, we propose a new model for freesquare random monomial ideals generated by edges and covers of a graph. This permit us to investigate the conditions of normality for which we obtain asymptotic results. We also elaborate on asymptotic results for other invariants such as the Krull dimension (for which we obtain threshold function), the regularity and the $v$-number.