Large random simplicial complexes, II; the fundamental group

Costa, A.; Farber, Michael

doi:10.1142/s1793525317500170

Cited by 24 publications

(32 citation statements)

References 28 publications

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“…In the present paper we study two very general probabilistic models generating random simplicial complexes of arbitrary dimension which we call the lower and upper models. random simplicial complexes was studied in a series of papers [2], [3], [4], [5] under the name of multiparameter random simplicial complexes; the name reflects the fact that the geometric and topological properties of simplicial complexes in this model depend on the set of probability parameters p 0 , p 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Random simplicial complexes, duality and the critical dimension

2019

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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 Betti numbers in the lower model is characterised by the notion of critical dimension, which was introduced by A. Costa and M. Farber in [5]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.In the case of the lower model one builds the random simplicial complex inductively, step by step, starting with a random set of vertices, then adding randomly edges between the selected vertices, and on the following step adding randomly 2-simplexes (triangles) to the random graph obtained on the previous stage, and so on. Examples are given by the Erdös -Rényi [7] random graphs and their high dimensional generalizations, the Linial, Meshulam, Wallach [14] , [15] random simplicial complexes, as well as random clique complexes [12], [13]. In larger generality the lower model of Michael

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Section: Introductionmentioning

confidence: 99%

Random simplicial complexes, duality and the critical dimension

2019

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“…Versions of this result for higher dimensional complexes are also developed in [8]. We also mention that our above threshold also happens to be the threshold for both uniform hyperbolicity and triviality of the fundamental group, as shown in [4].…”

Section: Introductionmentioning

confidence: 64%

“…For a 2-complex S let f i (S), i = 0, 1, 2, denote the number of simplices in S of dimension i (for us simplicial complexes are finite by definition). In this work we will apply Theorem 1(B) of [4] which states the following. If satisfies that for every subcomplex T of S…”

Section: Introductionmentioning

confidence: 99%

Topological embeddings into random 2‐complexes

Farber

Nowik

2021

Random Struct Algorithms

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We consider 2-dimensional random simplicial complexes Y in the multiparameter model. We establish the multiparameter threshold for the property that every 2-dimensional simplicial complex S admits a topological embedding into Y asymptotically almost surely. Namely, if in the procedure of the multiparameter model on n vertices, each i-dimensional simplex is taken with probability p i = p i (n), then the threshold is p 0 p 3 1 p 2 2 = 1 n . Our claim in one direction is in fact slightly stronger, namely, we show that if p 0 p 3 1 p 2 2 is sufficiently larger than 1 n then every S has a fixed subdivision S ′ which admits a simplicial embedding into Y asymptotically almost surely. In the other direction we show that if p 0 p 3 1 p 2 2 is sufficiently smaller than 1 n , then asymptotically almost surely, the torus does not admit a topological embedding into Y.

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“…Note that most random complexes which appear in literature are homogeneous. For example the multi-parameter random simplicial complexes of [6], [7], [8], [9] are homogeneous lower random simplicial complexes.…”

Section: 2mentioning

confidence: 99%

Random simplicial complexes in the medial regime

Farber

Mead

2020

Topology and its Applications

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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 pσ approach neither 0 nor 1. We show that nontrivial Betti numbers of typical lower and upper random simplicial complexes in the medial regime lie in a narrow range of dimensions. For instance, an upper random simplicial complex Y on n vertices in the medial regime with high probability has non-vanishing Betti numbers bj (Y ) only for k + c < n − j < k + log 2 k + c ′ where k = log 2 ln n and c, c ′ are constants. A lower random simplicial complex on n vertices in the medial regime is with high probability (k + a)connected and its dimension d satisfies d ∼ k + log 2 k + a ′ where a, a ′ are constants. The paper develops a new technique, based on Alexander duality, which relates the lower and upper models.

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Large random simplicial complexes, II; the fundamental group

Cited by 24 publications

References 28 publications

Random simplicial complexes, duality and the critical dimension

Random simplicial complexes, duality and the critical dimension

Topological embeddings into random 2‐complexes

Random simplicial complexes in the medial regime

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