论文标题
Erdös-rényi随机图的边缘理想：线性分辨率，无混合性和规律性
Edge ideals of Erdös-Rényi random graphs : Linear resolution, unmixedness and regularity
论文作者
论文摘要
我们研究Erdös-Rényi随机图的边缘理想的同源代数。这些随机图是通过在$ n $顶点上删除完整图的边缘，彼此独立于概率$ 1-p $而生成的。我们专注于这些随机边缘理想的某些方面 - 线性分辨率，未混合性和代数不变性，例如Castelnuovo -Mumford的规律性，投影维度和深度。我们首先显示了存在线性呈现和分辨率的双相转换，并也确定关键窗口。结果，除了非常具体的参数选择（即$ n，p：= p（n）$）外，我们获得了这一点，并且只有当它具有线性分辨率时，随机边缘理想具有线性呈现。这表明对于具有高概率的大型随机图的某些猜想是正确的，即使证明这些猜想因确定图形而失败。接下来，我们研究某些代数不变的渐近行为 - 稀疏状态中这种随机边缘理想的Castelnuovo -Mumford规律性，投影尺寸和深度 - $ P = \fracλ{n}，λ\，λ\ in（0，\ infty）$）。使用局部弱收敛（或Benjamini-Schramm收敛）研究这些不变性，并将它们与Galton-Watson树上的不变性联系起来。我们还表明，当$ p \ to 0 $或$ p \至1 $足够快时，那么较高的概率是毫无用处的，对于$ p $的大多数其他选择，这些理想并不具有很高的概率。这是朝着猜想的进一步发展，即在变量数量固定时，随机单独理想不太可能具有Cohen-Macaulay属性（请参见De Loera等人，2016年，2019b）。
We study the homological algebra of edge ideals of Erdös-Rényi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with probability $1-p$. We focus on some aspects of these random edge ideals - linear resolution, unmixedness and algebraic invariants like the Castelnuovo-Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., $n,p := p(n)$), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. Next, we study asymptotic behaviour of some algebraic invariants - the Castelnuovo-Mumford regularity, projective dimension and depth - of such random edge ideals in the sparse regime (i.e., $p = \fracλ{n}, λ\in (0,\infty)$). These invariants are studied using local weak convergence (or Benjamini-Schramm convergence) and relating them to invariants on Galton-Watson trees. We also show that when $p \to 0$ or $p \to 1$ fast enough, then with high probability the edge ideals are unmixed and for most other choices of $p$, these ideals are not unmixed with high probability. This is further progress towards the conjecture that random monomial ideals are unlikely to have Cohen-Macaulay property (see De Loera et al. 2019a,2019b) in the setting when the number of variables goes to infinity but the degree is fixed.
