The Threshold for Integer Homology in Random d-Complexes

@article{Hoffman2017TheTF,
  title={The Threshold for Integer Homology in Random d-Complexes},
  author={Christopher Hoffman and Matthew Kahle and Elliot Paquette},
  journal={Discrete \& Computational Geometry},
  year={2017},
  volume={57},
  pages={810-823}
}
Let $$Y \sim Y_d(n,p)$$Y∼Yd(n,p) denote the Bernoulli random d-dimensional simplicial complex. We answer a question of Linial and Meshulam from 2003, showing that the threshold for vanishing of homology $$H_{d-1}(Y; \mathbb {Z})$$Hd-1(Y;Z) is less than $$40d (d+1) \log n / n$$40d(d+1)logn/n. This bound is tight, up to a constant factor which depends on d. 
On simple connectivity of random 2-complexes
The fundamental group of the $2$-dimensional Linial-Meshulam random simplicial complex $Y_2(n,p)$ was first studied by Babson, Hoffman and Kahle. They proved that the threshold probability for simpleExpand
Small Simplicial Complexes with Prescribed Torsion in Homology
  • Andrew Newman
  • Mathematics, Computer Science
  • Discret. Comput. Geom.
  • 2019
TLDR
It is proved that for every d≥2 there exist constants c_d and C_d so that for any finite abelian group Td(G) which matches the known lower bound up to a constant factor. Expand
Topology and geometry of random 2-dimensional hypertrees
A hypertree, or $\mathbb{Q}$-acyclic complex, is a higher-dimensional analogue of a tree. We study random $2$-dimensional hypertrees according to the determinantal measure suggested by Lyons. We areExpand
Eigenvalue confinement and spectral gap for random simplicial complexes
TLDR
The main ingredient of the proof is a Furedi-Koml\'os-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries, and it is proved that the global distribution of the eigenvalues is asymptotically given by the semicircle law. Expand
Vanishing of cohomology groups of random simplicial complexes
TLDR
A hitting time result is proved, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction, and the asymptotic distribution of the dimension of the $j$-th cohomological group inside the critical window is studied. Expand
Integral Homology of Random Simplicial Complexes
TLDR
It is proved that with probability tending to 1 as narrowarrow n→∞, the first homology group over Z, Z vanishes at the very moment when all the edges are covered by triangular faces. Expand
The threshold for d-collapsibility in random complexes
TLDR
For every c>i¾?d, a complex drawn from Xdn,cn is asymptotically almost surely not d-collapsible, and here it is shown that this is indeed the correct threshold. Expand
Algebraic and combinatorial expansion in random simplicial complexes
In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a $d$-dimensional Linial-Meshulam random simplicial complex, above the cohomologicalExpand
Cohomology groups of non-uniform random simplicial complexes
We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial randomExpand
Spectral Gaps of Random Graphs and Applications
We study the spectral gap of the Erdős–Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta> 0$ if $$\begin{equation*} p \geq \frac{(1/2 +Expand
...
1
2
3
...

References

SHOWING 1-10 OF 20 REFERENCES
Homological connectivity of random k-dimensional complexes
TLDR
It is shown that p=(k \log n)/n is a sharp threshold for the vanishing of H_{k-1}(Y;R), the reduced homology group of Y with coefficients in a finite abelian group R. Expand
Sharp vanishing thresholds for cohomology of random flag complexes
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. WeExpand
The fundamental group of random 2-complexes
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 inExpand
Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes
TLDR
It is shown that there exists a constant $$\gamma _d< c_d c-d$$ and a fixed field $$\mathbb{F }$$, asymptotically almost surely $$H_d(Y;\mathBB{F }) \ne 0$$, and conjecture this bound to be sharp. Expand
Topology of Random 2-Complexes
TLDR
This paper studies the Linial–Meshulam model of random two-dimensional simplicial complexes and proves that for p≪n−1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π1(Y) is free and H2(Y)=0, asymptotically almost surely. Expand
Homological Connectivity Of Random 2-Complexes
TLDR
It is shown that for any function ω(n) that tends to infinity, H_{1) is the first homology group of Y with mod 2 coefficients. Expand
Spectral Gaps of Random Graphs and Applications
We study the spectral gap of the Erdős–Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta> 0$ if $$\begin{equation*} p \geq \frac{(1/2 +Expand
Enumeration ofQ-acyclic simplicial complexes
Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely (kn−1)k-faces, (3)Hk(C)=0. We prove thatExpand
A random graph
Abstract : Let X(1),X(2), ..., X(n) be independent random variables such that P(X(i) = j) = P sub j , j = 1,2, ..., n, sum from j = 1 to n of P sub j = 1 and consider a graph with n nodes numberedExpand
Perfect forms and the Vandiver conjecture
Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that theExpand
...
1
2
...