We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {mathbb K}[x_1, dots x_n]$, utilizing methods from the Erd{o}s-R{e}nyi model of random graphs. Here for a graph $G sim G(n, p)$ we consider the `coedge idea
l $I_G$ corresponding to the missing edges of $G$, and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results involve fixing the edge probability $p = p(n)$ so that asymptotically almost surely the Krull dimension of $R/I_G$ is fixed. Under these conditions we establish various properties regarding the Betti table of $R/I_G$, including sharp bounds on regularity and projective dimension, and distribution of nonzero normalized Betti numbers. These results extend work of Erman and Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex $k$-connectivity that may be of independent interest.
We define a new family of graph invariants, studying the topology of the moduli space of their geometric realizations in Euclidean spaces, using a limiting procedure reminiscent of Floer homology. Given a labeled graph $G$ on $n$ vertices and $d geq
1$, $W_{G, d} subseteq mathbb{R}^{d times n}$ denotes the space of nondegenerate realizations of $G$ in $mathbb{R}^d$.The set $W_{G, d}$ might not be connected, even when it is nonempty, and we refer to its connected components as rigid isotopy classes of $G$ in $mathbb{R}^d$. We study the topology of these rigid isotopy classes. First, regarding the connectivity of $W_{G, d}$, we generalize a result of Maehara that $W_{G, d}$ is nonempty for $d geq n$ to show that $W_{G, d}$ is $k$-connected for $d geq n + k + 1$, and so $W_{G, infty}$ is always contractible. While $pi_k(W_{G, d}) = 0$ for $G$, $k$ fixed and $d$ large enough, we also prove that, in spite of this, when $dto infty$ the structure of the nonvanishing homology of $W_{G, d}$ exhibits a stabilization phenomenon: it consists of $(n-1)$ equally spaced clusters whose shape does not depend on $d$, for $d$ large enough. This leads to the definition of a family of graph invariants, capturing this structure. For instance, the sum of the Betti numbers of $W_{G,d}$ does not depend on $d$, for $d$ large enough; we call this number the Floer number of the graph $G$. Finally, we give asymptotic estimates on the number of rigid isotopy classes of $mathbb{R}^d$--geometric graphs on $n$ vertices for $d$ fixed and $n$ tending to infinity. When $d=1$ we show that asymptotically as $nto infty$ each isomorphism class corresponds to a constant number of rigid isotopy classes, on average. For $d>1$ we prove a similar statement at the logarithmic scale.
A two-step model for generating random polytopes is considered. For parameters $d$, $m$, and $p$, the first step is to generate a simple polytope $P$ whose facets are given by $m$ uniform random hyperplanes tangent to the unit sphere in $mathbb{R}^d$
, and the second step is to sample each vertex of $P$ independently with probability $p$ and let $Q$ be the convex hull of the sampled vertices. We establish results on how well $Q$ approximates the unit sphere in terms of $m$ and $p$ as well as asymptotics on the combinatorial complexity of $Q$ for certain regimes of $p$.
We consider the question of the largest possible combinatorial diameter among $(d-1)$-dimensional simplicial complexes on $n$ vertices, denoted $H_s(n, d)$. Using a probabilistic construction we give a new lower bound on $H_s(n, d)$ that is within an
$O(d^2)$ factor of the upper bound. This improves on the previously best-known lower bound which was within a factor of $e^{Theta(d)}$ of the upper bound. We also make a similar improvement in the case of pseudomanifolds.
