Edge ideals of Erdős–Rényi  random graphs: linear resolution, unmixedness and regularity

We study the homological algebra of edge ideals of Erdős–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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-p$$\end{document}. 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n,p:= p(n)$$\end{document}), 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=λn,λ∈(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p = \frac{\lambda }{n}, \lambda \in (0,\infty )$$\end{document}). 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→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \rightarrow 0$$\end{document} or p→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \rightarrow 1$$\end{document} 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 (De Loera et al. in Proc Am Math Soc 147(8):3239–3257, 2019; J Algebra 519:440–473, 2019) in the setting when the number of variables goes to infinity but the degree is fixed.