# Random subcomplexes and Betti numbers of random edge ideals

@inproceedings{Dochtermann2021RandomSA, title={Random subcomplexes and Betti numbers of random edge ideals}, author={Anton Dochtermann and Andrew Newman}, year={2021} }

We study homological properties of random quadratic monomial ideals in a polynomial ring R = K[x1, . . . xn], utilizing methods from the Erdős–Rényi model of random graphs. Here for a graph G ∼ G(n,p) we consider the ‘coedge’ ideal IG corresponding to the missing edges of G, and study Betti numbers of R/IG as n tends to infinity. Our main results involve fixing the edge probability p = p(n) so that asymptotically almost surely the Krull dimension of R/IG is fixed. Under these conditions we…

## One Citation

### The regularity of almost all edge ideals

- Mathematics
- 2021

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The “almost” is crucial, without it there is…

## References

SHOWING 1-10 OF 39 REFERENCES

### Extremal Betti Numbers and Applications to Monomial Ideals

- Mathematics
- 1998

Recall that the (Mumford-Castelnuovo) regularity of M is the least integer ρ such that for each i all free generators of Fi lie in degree ≤ i + ρ, that is βi,j = 0, for j > i + ρ. In terms of…

### Average behavior of minimal free resolutions of monomial ideals

- MathematicsProceedings of the American Mathematical Society
- 2019

We show that, under a natural probability distribution, random monomial ideals will almost always have minimal free resolutions of maximal length; that is, the projective dimension will almost always…

### Edge ideals of Erd\"{o}s-R\'enyi random graphs : Linear resolution, unmixedness and regularity.

- Mathematics
- 2020

We study the homological algebra of edge ideals of Erdos-Renyi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with…

### Asymptotics of random Betti tables

- Mathematics
- 2015

The purpose of this paper is twofold. First, we present a conjecture to the effect that the ranks of the syzygy modules of a smooth projective variety become normally distributed as the positivity of…

### Threshold functions for small subgraphs

- Mathematics, Computer ScienceMathematical Proceedings of the Cambridge Philosophical Society
- 1981

In this note, random labelled graphs are studied by the set of all graphs with n given labelled vertices and M(n) edges by turning (M(n)) into a probability space by giving all graphs the same probability.

### Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutions

- Mathematics
- 2019

In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide…

### On the phase transition in random simplicial complexes

- Mathematics
- 2014

It is well-known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph is, almost surely, no longer a forest, and here it first…

### Extremal Betti numbers of edge ideals

- MathematicsArchiv der Mathematik
- 2019

Given integers r and b with $$1 \le b \le r$$1≤b≤r, a finite simple connected graph G for which $$\mathrm{reg}(S/I(G)) = r$$reg(S/I(G))=r and the number of extremal Betti numbers of S / I(G) is equal…