The Threshold for Integer Homology in Random d-Complexes

@article{Hoffman2013TheTF,
  title={The Threshold for Integer Homology in Random d-Complexes},
  author={Christopher Hoffman and Matthew Kahle and Elliot Paquette},
  journal={Discrete \& Computational Geometry},
  year={2013},
  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. 
View on Springer
arxiv.org
35 Citations
Highly Influential Citations
3
Background Citations
16
Methods Citations
4
Results Citations
1

35 Citations

The integer homology threshold in $Y_d(n, p)$

We prove that in the $d$-dimensional Linial--Meshulam stochastic process the $(d - 1)$st homology group with integer coefficients vanishes exactly when the final isolated $(d - 1)$-dimensional face…

On Simple Connectivity of Random 2-Complexes

This paper shows that p = ( γ n ) - 1 / 2 is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of Y 2 ( n, p) that is homeomorphic to a disk.

Small Simplicial Complexes with Prescribed Torsion in Homology

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.

Eigenvalue confinement and spectral gap for random simplicial complexes

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.

Integral Homology of Random Simplicial Complexes

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.

The threshold for d‐collapsibility in random complexes*

For every c > ηd, a complex drawn from Xd(n,cn) is asymptotically almost surely not d‐collapsible, and a lower bound for this threshold p=ηdn was established in (Aronshtam and Linial, Random Struct.

Eigenvalues and spectral gap in sparse random simplicial complexes

We consider the adjacency operator A of the Linial-Meshulam model X(d, n, p) for random d−dimensional simplicial complexes on n vertices, where each d−cell is added independently with probability p ∈…

The homology of random simplicial complexes in the multi-parameter upper model

. We study random simplicial complexes in the multi-parameter upper model. In this model simplices of various dimensions are taken randomly and independently, and our random simplicial complex Y is…

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 +…

Phase Transition in 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 random…

References

SHOWING 1-10 OF 21 REFERENCES

Homological connectivity of random k‐dimensional complexes

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.

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. We…

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

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.

Topology of Random 2-Complexes

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.

Homological Connectivity Of Random 2-Complexes

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.

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 +…

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 that…

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 the…

Sharp thresholds of graph properties, and the -sat problem

Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone…

On L2-cohomology and Property (T) for Automorphism Groups of Polyhedral Cell Complexes

Abstract. We present an update of Garland's work on the cohomology of certain groups, construct a class of groups many of which satisfy Kazhdan's Property (T) and show that properly discontinuous and…