Erratum: Limit theorems for Betti numbers of random simplicial complexes
@article{Kahle2015ErratumLT, title={Erratum: Limit theorems for Betti numbers of random simplicial complexes}, author={Matthew Kahle and Elizabeth S. Meckes}, journal={arXiv: Probability}, year={2015} }
We correct the proofs of the main theorems in our paper "Limit theorems for Betti numbers of random simplicial complexes".
8 Citations
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References
SHOWING 1-10 OF 21 REFERENCES
Limit theorems for Betti numbers of random simplicial complexes
- Mathematics
- 2010
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases…
Algebraic Topology
- Mathematics
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
Random graphs
- Mathematics, Computer ScienceSODA '06
- 2006
Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Random Geometric Complexes
- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 2011
The expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd are studied and asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes are given.
A central limit theorem for decomposable random variables with applications to random graphs
- MathematicsJ. Comb. Theory, Ser. B
- 1989
The fundamental group of random 2-complexes
- Mathematics
- 2007
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in…
Sharp vanishing thresholds for cohomology of random flag complexes
- Mathematics
- 2012
For every $k \ge 1$, the $k$th cohomology group $H^k(X, \Q)$ of the random flag complex $X \sim X(n,p)$ passes through two phase transitions: one where it appears, and one where it vanishes. We…
Random Geometric Graphs
- Mathematics
- 2003
This chapter discusses probabilistic ingredients, the largest component for a binomial process, and connectedivity and the number of components in a graph-like model.