[论文解读] Probabilities of random monomial ideals associated to large graphs

Daniel George, Humberto Muñoz-George+1 more|arXiv (Cornell University)|Jan 11, 2026

Geometry and complex manifolds被引用 0

一句话总结

该论文提出一种基于 Erdős–Rényi 图的随机边和覆盖单纯理想的概率模型，并分析渐近正态性、Krull 维数、正则性与 v-number。它为决定这些代数不变量的图属性阈值函数提供了阈值函数。

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.

研究动机与目标

• Motivate the use of randomness to understand average behavior of monomial ideals associated to graphs.
• Introduce Erdős–Rényi inspired models for edge and cover ideals of graphs and study when these ideals are normal.
• Determine threshold functions for Krull dimension and derive asymptotics for regularity and v-number.

提出的方法

• Define random graphs G with vertex set corresponding to variables and edges chosen with probability p.
• Define edge ideal I(G) and cover ideal Ic(G) from G as squarefree monomial ideals.
• Prove asymptotic normality criteria for I(G) and Ic(G) under conditions on p, q=1-p.
• Relate Krull dimension to the independence number β0(G) and derive threshold functions for dim(S/I(G)).
• Use Hochster configurations and duality criteria to connect normality of Ic(G) to properties of G and its complement.
• Employ probabilistic counting for induced subgraphs like T and Et to estimate events driving normality and dimension.

实验结果

研究问题

• RQ1 What are the asymptotic probabilities that I(G) and Ic(G) are normal under Erdős–Rényi distributions?
• RQ2 How does the Krull dimension of S/I(G) behave asymptotically, and what are the threshold functions for β0(G)≥t?
• RQ3 What are the asymptotics for the regularity and v-number of these random monomial ideals under varying p?
• RQ4 How do properties of the graph, such as Hochster configurations, influence normality via duality?

主要发现

• For I(G) ~ IER(n,p), normality occurs with probability approaching 1 when p = o(1/n) and tends to 0 when pq^{3/2} = ω(1/n).
• For Ic(G) ~ IERc(n,p), normality holds with probability 1 if p = o(1/n).
• If q = 1-p = o(1/n), then Ic(G) is not normal with β0(G)≤2 has probability approaching 0.
• The Krull dimension of S/I(G) equals the independence number β0(G), with a threshold q ~ n^{-2/(t-1)} for dim(S/I(G))≥t; thus dim equals t asymptotically almost surely when q lies between o(n^{-2/t}) and ω(n^{-2/(t-1)}).
• Corollaries give reg(S/I(G)) ≤ t-1 and v(I(G)) ≤ t-1 under q = o(n^{-2/(t-1)}).