Related papers: The threshold for integer homology in random d-com…

Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by Htr D, is defined to be the injective dimension of the enveloping algebra of D. We show that Htr has several useful properties which the…

Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which…

Consider a random multigraph G* with given vertex degrees d_1,...,d_n, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending to infinity, the…

We study random 2-dimensional complexes in the Linial - Meshulam model and prove that for the probability parameter satisfying $$p\ll n^{-46/47}$$ a random 2-complex $Y$ contains several pairwise disjoint tetrahedra such that the 2-complex…

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all…

In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties…

A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard…

It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In…

Given a Poisson process on a $d$-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the \u{C}ech complex associated to the coverage of…

While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising…

We consider the Erd\H{o}s-R\'enyi evolution of random graphs, where a new uniformly distributed edge is added to the graph in every step. For every fixed $d\ge 1$, we show that with high probability, the graph becomes rigid in $\mathbb R^d$…

We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0,1]-valued…

We introduce several notions of `random fewnomials', i.e. random polynomials with a fixed number f of monomials of degree N. The f exponents are chosen at random and then the coefficients are chosen to be Gaussian random, mainly from the…

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$…

We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\displaystyle D(G)=\max_{H\subseteq G}\delta(H)$. This is a strong generalization of a theorem of Cuki\'{c} and Kozlov,…

This paper studies the homology and cohomology of the Temperley-Lieb algebra TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a=v+v^{-1} for some unit v in the ground ring, and…

For each fixed $d\ge 1$, we obtain asymptotic estimates for the number of $d$-representable simplicial complexes on $n$ vertices as a function of $n$. The case $d=1$ corresponds to counting interval graphs, and we obtain new results in this…

Let $ \mathbb{L}^{d} = ( \mathbb{Z}^{d},\mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ \mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional…

We perform Transition matrix Monte Carlo simulations to evaluate the entropy of rhombus tilings with fixed polygonal boundaries and 2D-fold rotational symmetry. We estimate the large-size limit of this entropy for D=4 to 10. We confirm…

Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic…