Related papers: The homology of random simplicial complexes in the…
We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or…
The random $2$-dimensional simplicial complex process starts with a complete graph on $n$ vertices, and in every step a new $2$-dimensional face, chosen uniformly at random, is added. We prove that with probability tending to $1$ as…
We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters $p_\sigma$ approach neither $0$ nor $1$. We show that nontrivial Betti numbers…
Let Delta_{n-1} denote the (n-1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of Delta_{n-1} obtained by starting with the full (k-1)-dimensional skeleton of Delta_{n-1} and then adding each k-simplex independently with…
We consider a multi-parameter model for randomly constructing simplicial complexes. This model interpolates between random clique complexes and Linial-Meshulam random $k$-dimensional complexes, two models that have been extensively studied.…
The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate…
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…
In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special…
We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large…
Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and…
Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. We compute an explicit…
We consider 2-dimensional random simplicial complexes $Y$ in the multi-parameter model. We establish the multi-parameter threshold for the property that every 2-dimensional simplicial complex $S$ admits a topological embedding into $Y$…
In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the…
There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a…
Several years ago Linial and Meshulam introduced a model called X_d(n,p) of random n-vertex d-dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability p=p(n) at which the…
In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random…
We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first…
In this paper we study the Linial-Meshulam model of random two-dimensional complexes. We prove that a random 2-complex is homotopically one dimensional, with probability tending to one as n tends to infitnity, assuming that the probability…
The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…