Probabilities of random monomial ideals associated to large graphs

TL;DR

This work introduces a probabilistic model for random monomial ideals generated by graph edges and vertex covers using a random graph $Gigl( ER(n,p) igr)$. It connects algebraic properties of the associated ideals $I(G)$ and $I_c(G)$ to graph-theoretic invariants, establishing threshold functions for the Krull dimension and to asymptotically determine normality, while also deriving asymptotic bounds for regularity and the $v$-number. The main contributions include precise conditions under which $I(G)$ and $I_c(G)$ are normal, a threshold for $ ext{dim}(S/I(G))$ in terms of induced empty subgraphs, and corollaries bounding homological invariants, all derived through Hochster configurations, induced-subgraph counts, and duality arguments. The results highlight a deep interplay between random graph theory and the algebraic behavior of squarefree monomial ideals and provide asymptotic characterizations for large graphs that inform both theory and potential applications in combinatorial algebraic geometry.

Abstract

Inspired by the Erdős Rényi 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.

Figures (6)

• Figure 1: The graph of example \ref{['Example: ideal of edges and convers']}.
• Figure 2: A Hochster configuration.
• Figure 3: The graph $C_5$ (left) and its complement $\overline{G}$ (right).
• Figure 4: A graph $G$ (left) and an induced subgraph of $G$ (right). Here $V^\prime=\{x_1,x_2,x_4,x_5,x_6\}$. Oberve that the graph on the right is also the restriction of $G$ to $V^\prime$, $G_{V^\prime}$.
• Figure 5: The graph $T$ (left) and the graph $E_t$ (right) with $t=6$.

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 2.7
• Theorem 2.8
• Corollary 2.9
• Proposition 2.10