Freesquare Random Monomial Ideals

• Freesquare random monomial ideals are probabilistic constructs defined by squarefree generators from Erdős–Rényi graphs, linking combinatorial algebra with random graph theory.
• They use probabilistic thresholds to characterize key invariants such as normality and Krull dimension by relating specific graph substructures to algebraic properties.
• The model connects Castelnuovo–Mumford regularity and the v-number to independent set sizes, offering insights into asymptotic behaviors in both algebra and graph theory.

Freesquare random monomial ideals are probabilistic objects arising from the interplay between combinatorial commutative algebra and random graph theory. The model provides a rigorous framework for studying the asymptotic algebraic properties of squarefree monomial ideals defined in terms of the edge and cover structures of Erdős–Rényi random graphs. Key invariants such as normality, Krull dimension, Castelnuovo–Mumford regularity, and the $v$-number exhibit sharp threshold phenomena reflecting the underlying graph structure, translating classical random graph thresholds into the language of commutative algebra (George et al., 11 Jan 2026).

1. Random Graph-Induced Monomial Ideals

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$. Given a random graph $G=(V,E)$ on the vertex set $V=\{x_1,\ldots,x_n\}$ sampled from the Erdős–Rényi model $\mathrm{ER}(n,p)$, where each edge $\{x_i,x_j\}$ ($i<j$) occurs independently with probability $p$ ($q=1-p$), two canonical squarefree (freesquare) monomial ideals in $S$ are defined:

• Edge ideal: $I(G) = (x_ix_j~:~\{x_i,x_j\}\in E)$.
• Cover ideal: $$I_c(G) = (x_{i_1}\cdots x_{i_r}~:~\{x_{i_1},\ldots,x_{i_r}\}$$ is a minimal vertex cover of $G$).

The law $I(G)\sim \mathrm{IER}(n,p)$ describes a random edge ideal; $I_c(G)\sim \mathrm{IER}_c(n,p)$ for the random cover ideal. The probability of obtaining a fixed edge ideal supported on $m$ edges is $p^m q^{\binom{n}{2} - m}$. These constructs are termed "freesquare" random monomial ideals, in direct reference to their squarefree generators (George et al., 11 Jan 2026).

2. Normality: Criteria and Asymptotics

A monomial ideal $I\subset S$ is said to be normal if $I^k$ is integrally closed for every $k\ge1$. Graph-theoretically, normality for edge and cover ideals corresponds to the absence of certain substructures, known as Hochster configurations (pairs of vertex-disjoint induced odd cycles):

• $I(G)$ is normal if and only if $G$ lacks a Hochster configuration (per Gitler–Reyes–Villarreal; Simis–Vasconcelos–Villarreal).
• $I_c(G)$ is normal if and only if the complement $\overline{G}$ lacks a Hochster configuration, provided that the independence number $\beta_0(G)\le2$ (per Dupont–Muñoz–Villarreal).

Thresholds for normality in the asymptotic regime ($n\to\infty$) are given by:

• For $I(G)\sim \mathrm{IER}(n,p)$,
  • $P[I(G)\text{ normal}]\to1$ if $p=o(1/n)$,
  • $P[I(G)\text{ normal}]\to0$ if $p q^{3/2} = \omega(1/n)$.
• For $I_c(G)\sim \mathrm{IER}_c(n,p)$,
  • If $p=o(1/n)$, then $P[I_c(G)\text{ normal}]\to1$,
  • If $q=o(1/n)$, then $$P[I_c(G)\text{ not normal}\land \beta_0(G)\le2]\to0$$.

The sharpness of these thresholds is established via probabilistic bounds on the emergence of Hochster configurations, particularly using expectations and variances of induced subgraphs such as disjoint triangles. The normality criteria directly mirror classical results on cycle and subgraph appearance in Erdős–Rényi random graphs (George et al., 11 Jan 2026).

3. Krull Dimension and Threshold Phenomena

For a freesquare edge ideal $I(G)$, the Krull dimension of the quotient ring coincides with the size of the largest independent set $\beta_0(G)$ in $G$: $\dim S/I(G)=\beta_0(G)$. The existence of large independent sets, and hence large Krull dimension, is governed by the appearance of empty induced subgraphs.

Let $E_t$ denote the empty graph on $t$ vertices. The threshold for the event $\{\beta_0(G)\ge t\}$, equivalently $\{\dim S/I(G)\ge t\}$, is given by $q=1/n^{2/(t-1)}$:

• $P[\dim S/I(G)\ge t]\to 0$ if $q=o(1/n^{2/(t-1)})$,
• $P[\dim S/I(G)\ge t]\to 1$ if $q=\omega(1/n^{2/(t-1)})$.

For fixed $t$, more refined results yield that $P[\dim S/I(G)=t]\to1$ if $q=o(1/n^{2/t})$ and $q=\omega(1/n^{2/(t-1)})$. These threshold functions are derived using subgraph-counting random variables and probabilistic inequalities (Markov, Chebyshev) applied to $Y_{E_t}$, the number of induced $t$-vertex empty graphs (George et al., 11 Jan 2026).

4. Regularity and $v$-Number Asymptotics

The Castelnuovo–Mumford regularity $\mathrm{reg}(S/I)$ and the $v$-number $v(I)$ are critical algebraic invariants for monomial ideals. For squarefree edge ideals, the following combinatorial-algebraic bounds (Delio–Villarreal) hold:

• $\mathrm{reg}(S/I(G))\le \dim S/I(G)$,
• $v(I(G))\le \beta_0(G) = \dim S/I(G)$.

These yield immediate asymptotic consequences. Under $q=o(1/n^{2/(t-1)})$:

• $P[\mathrm{reg}(S/I(G))\le t-1]\to1$,
• $P[v(I(G))\le t-1]\to1$.

Thus, as the edge probability decreases, both regularity and $v$-number of random edge ideals become sharply bounded above, inheriting the threshold behavior from the Krull dimension (George et al., 11 Jan 2026).

5. Probabilistic Tools and Subgraph Containment

The analysis employs random variables $Y_T(n,p)$ and $Y_{E_t}(n,p)$, counting the number of induced copies of a fixed graph $T$ or empty $t$-vertex subgraphs, respectively. Expectations and variances for these counts are central to deriving thresholds for the appearance of graph-theoretical features that determine corresponding algebraic properties.

Classical phenomena from random graph theory reappear:

• The threshold for the appearance of a fixed graph $H$ as an induced subgraph is at $p\approx n^{-1/\Delta(H)}$, where $\Delta(H)$ is the maximum degree of $H$.
• The random graph is acyclic (contains no cycles) asymptotically almost surely when $p=o(1/n)$.

The algebraic results for freesquare random monomial ideals are thus direct translations of these random graph thresholds within commutative algebra, linking probabilistic combinatorics and algebraic invariants (George et al., 11 Jan 2026).

6. Significance and Interplay with Graph Theory

The freesquare random monomial ideal model establishes a fertile bridge between algebraic invariants and random combinatorics. Asymptotic properties such as normality, Krull dimension, regularity, and $v$-number are governed not only by algebraic considerations but are intimately controlled by the emergence or absence of graph-theoretic substructures in the underlying Erdős–Rényi process.

The equivalences and threshold results demonstrate that key commutative algebraic phenomena, such as the integrality of powers or the complexity of syzygies encoded by regularity, can be probabilistically characterized in terms of the phase transitions in random graphs. This framework unifies and extends both algebraic statistics and random graph theory by expressing algebraic invariants as probabilistic events on random structures (George et al., 11 Jan 2026).