Homological Connectivity Of Random 2-Complexes

Linial, Nathan; Meshulam, Roy

doi:10.1007/s00493-006-0027-9

Cited by 262 publications

(380 citation statements)

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“…In recent years, there have been interesting developments in the study of random simplicial complexes (Linial and Meshulam 2006;Adler et al 2010;Kahle 2016), random groups (Gromov 2003;Ollivier 2005), random manifolds (Brooks et al 2004;Dunfield and Thurston 2006;Pippenger and Schleich 2006;Farber and Kappeler 2008), and more.…”

Section: Random 3-manifoldsmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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Section: Random 3-manifoldsmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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“…Fix such a u ∈ F 1,c (W ). For a face t ∈ F 2 (W ) to contribute more than −f t u to the sum in (10) or (11) …”

Section: Lemma 411 If W Is a 2-dimensional Stratified Complex So Thmentioning

confidence: 99%

“…If H 1 (X, Z/pZ) = 0 for every p, then H 1 (X, Z) = 0 as well [7]. By the Linial-Meshulam-Wallach results ( [10,11]), H 1 (Y, Z) is finite and has no p-torsion for any fixed p. So once p 2 log n/n, either H 1 (Y, Z) is trivial, or it is a finite generated abelian group with torsion approaching infinity. The first scenario might seem more plausible, but as far as we know, nothing is proved either way.…”

Section: Open Problemsmentioning

confidence: 99%

The fundamental group of random 2-complexes

Babson

Hoffman

Kahle

2010

J. Amer. Math. Soc.

160

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We study Linial-Meshulam random 2 2 -complexes Y ( n , p ) Y(n,p) , which are 2 2 -dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n − 1 / 2 p = n^{-1/2} . This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log ⁡ n / n p = 2 \log n / n . We use a variant of Gromov’s local-to-global theorem for linear isoperimetric inequalities to show that when p = O ( n − 1 / 2 − ϵ p = O( n^{-1/2 -\epsilon } ), the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2 2 -dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.

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“…More recently, the two-dimensional analog Y n,p,2 of the Erdős-Rényi model was considered by Linial-Meshulam in [11], and, further, the d-dimensional model Y n,p,d , for d ≥ 3, was considered by Meshulam-Wallach in [13].…”

Section: Theorem 11 (Erdős-rényi Theorem [4]) Assume That W(n) Is mentioning

confidence: 99%

“…Theorem 1.1 (Erdős-Rényi Theorem, [4]). Assume that w(n) is any function w : N → R, such that lim n→∞ w(n) = ∞, and p = p(n) is the probability depending on n. Then we have More recently, the two-dimensional analog Y n,p,2 of the Erdős-Rényi model was considered by Linial-Meshulam in [11], and, further, the d-dimensional model Y n,p,d , for d ≥ 3, was considered by Meshulam-Wallach in [13].…”

Section: Thresholds For Vanishing Of the (D − 1)st Homology Group Of mentioning

confidence: 99%

The threshold function for vanishing of the top homology group of random 𝑑-complexes

Kozlov

2010

Proc. Amer. Math. Soc.

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Abstract. For positive integers n and d, and the probability function 0 ≤ p(n) ≤ 1, we let Y n,p,d denote the probability space of all at most d-dimensional simplicial complexes on n vertices, which contain the full (d − 1)-dimensional skeleton, and whose d-simplices appear with probability p(n). In this paper we determine the threshold function for vanishing of the top homology group in Y n,p,d , for all d ≥ 1. Thresholds for vanishing of the (d − 1)st homology group of random d-complexesIn 1959 Erdős and Rényi defined a natural model for random graphs which has since become classical. In this model, which we call 1 Y n,p,1 , the random graph has n vertices, and the edges are chosen uniformly and independently at random with probability p. Usually, one is interested in questions concerning various statistics on this probability space, in the situation when n goes to infinity, and p is a function of n. One of the main results of Erdős-Rényi concerning Y n,p,1 was the discovery of the threshold function for the connectivity of the graph. More precisely, reformulated in our language, they have shown the following theorem. Theorem 1.1 (Erdős-Rényi Theorem, [4]). Assume that w(n) is any functionw : N → R, such that lim n→∞ w(n) = ∞, and p = p(n) is the probability depending on n. Then we haveMore recently, the two-dimensional analog Y n,p,2 of the Erdős-Rényi model was considered by Linial-Meshulam in [11], and, further, the d-dimensional model Y n,p,d , for d ≥ 3, was considered by Meshulam-Wallach in [13].

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Homological Connectivity Of Random 2-Complexes

Cited by 262 publications

References 2 publications

Models of random knots

Models of random knots

The fundamental group of random 2-complexes

The threshold function for vanishing of the top homology group of random 𝑑-complexes

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