Related papers: Random simplicial complexes
We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its…
We show that distinct emergent symmetries, such as partial dynamical symmetry and quasi dynamical symmetry, can occur simultaneously in the same or different eigenstates of the Hamiltonian. Implications for nuclear spectroscopy in the…
We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either…
We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For…
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 review topological properties of K\"ahler and symplectic manifolds, and of their odd-dimensional counterparts, coK\"ahler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also…
Let X_R be the zero locus in RP^n of one or two independently and Weyl distributed random real quadratic forms (this is the same as requiring that the corresponding symmetric matrices are in the Gaussian Orthogonal Ensemble). We prove that…
We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps uniformly over $\mathbb{F}_q$, conditioned on ensuring that…
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…
We consider a variety of connections between threshold graphs, shifted complexes, and simplicial complexes naturally formed from a graph. These graphical complexes include the independent set, neighborhood, and dominance complexes. We…
A thorough discussion of the statistical ensemble of scale-free connected random tree graphs is presented. Methods borrowed from field theory are used to define the ensemble and to study analytically its properties. The ensemble is…
Without leaving finite mathematics and using finite topological spaces only, we give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs. Besides exploring the definition in various contexts, we add some…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…
Topological properties of solid states have sparked considerable recent interest due to their importance in the physics of lattices with a non-trivial basis and their potential in the design of novel materials. Here we describe an…
We provide dual algorithms for sampling the space of abstract simplicial complexes on a fixed number of vertices. We develop a generative and descriptive sampler designed with heuristics to help balance the combinatorial multiplicities of…
It has been known for years how random height variations of a repeated nano-scale structure can give rise to smooth angular color variations instead of the well-known diffraction pattern experienced if no randomization is present. However,…
We study high dimensional expansion beyond simplicial complexes (posets) and focus on $q$-complexes which are complexes whose basic building blocks are linear spaces. We show that the complete $q$-complex (consists of all subspaces of a…
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…
We investigate the geometry of a quantum universe with the topology of the four-torus. The study of non-contractible geodesic loops reveals that a typical quantum geometry consists of a small semi-classical toroidal bulk part, dressed with…