Techniques and outline

The aforementioned work of Andriamampianina and Ravelomanana [2005], Karoński and Łuczak [1997, 2002] on the giant component for random hypergraphs relies on enumer-ative techniques to a significant extent; for the basis Andriamampianina and Ravelo-manana [2005], Karoński and Łuczak [1997, 2002] are results on the asymptotic number of connected hypergraphs with a given number of vertices and edges. By contrast, in the present work we employ neither enumerative techniques nor results, but rely solely on probabilistic methods. Our proof methods are also quite different from Stepanov [1970], who first estimates the asymptotic probability that a random graph G_n,p is connected

1.2. Techniques and outline.

in order to determine the distribution of N(Gn,p). By contrast, in the present work we prove the local limit theorem forN(H_d(n, p)) directly, thereby obtaining “en passant” a new proof for the local limit theorem for random graphsG_n,p, which may be of indepen-dent interest. Besides, the local limit theorem can be used to compute the asymptotic probability thatGn,por, more generally,H_d(n, p) is connected, or to compute the asymp-totic number of connected hypergraphs with a given number of vertices and edges (cf.

Chapter 5). Hence, the general approach taken in the present work is actually converse to the prior ones Andriamampianina and Ravelomanana [2005], Karoński and Łuczak [1997, 2002], Stepanov [1970].

The proof of Theorem 2.1 makes use ofStein’s method, which is a general technique for proving central limit theorems (Stein [1970]). Roughly speaking, Stein’s result implies that a sum of a family of dependent random variables converges to the normal distribution if one can bound the correlations within any constant-sized subfamily sufficiently well.

The method was used by Barbour et al. [1989] in order to prove that in a random graph G_n,p, e.g., the number of tree components of a given (bounded) size is asymptotically normal. To establish Theorem 2.1, we extend their techniques in two ways.

• Instead of dealing with the number of vertices in trees of a given size, we apply Stein’s method to thetotal numbern− N(Hd(n, p)) of vertices outside of the giant component; this essentially means that we need to sum over all possible tree sizes up to about lnn.

• Since we are dealing with hypergraphs rather than graphs, we are facing a somewhat more complex situation than Barbour et al. [1989], because the fact that an edge may involve an arbitrary number dof vertices yields additional dependencies.

The main contribution of the first part of this thesis is the proof of Theorem 3.1.

To this end, we think of the edges of H_d(n, p) as being added in two “portions”. More precisely, we first include each possible edge with probability p₁ = (1−ε)p indepen-dently, where ε > 0 is small but independent of n (and denote the resulting random hypergraph byH₁); by Theorem 2.1, the orderN(H₁) of the largest component ofH₁ is asymptotically normal. Then, we add each possible edge that is not present in H₁ with a small probability p2 ∼ εp and investigate closely how these additional random edges attach further vertices to the largest component of H1. Denoting the number of these

“attached” vertices by S, we will show that the conditional distribution of S given the value ofN(H1) satisfies a local limit theorem. Sincep1 andp2 are chosen such that each edge is present with probabilityp after the second portion of edges has been added, this yields the desired result onN(H_d(n, p)).

The analysis of the conditional distribution of S involves proving that S is asymp-totically normal. To show this, we employ Stein’s method once more. In addition, in order to show thatS satisfies alocal limit theorem, we prove that the number of isolated vertices ofH1 that get attached to the largest component ofH1 by the second portion of random edges is binomially distributed. Since the binomial distribution satisfies a local limit theorem, we thus obtain a local limit theorem forS.

Our proof of Theorem 3.1 makes use of some results on the component structure of H_d(n, p) derived in Coja-Oghlan et al. [2006]. For instance, we employ the results on the expectation and the variance ofN(H_d(n, p)) from that paper. Furthermore, the analysis ofS given in the present work is a considerable extension of the argument used in Coja-Oghlan et al. [2006] in order to estimate the probability thatH_d(n, p) is connected up to a constant factor.

To prove Theorems 4.1 and 4.3, we build upon a qualitative result on the connected components of Hd(n, p) from Coja-Oghlan et al. [2006] (Theorems 1.2 and 3.1, cf. Sec-tion 1.3). The proofs of these ingredients solely rely on probabilistic reasoning (namely, branching processes and Stein’s method for proving convergence to a Gaussian).

In Section 4.2 we show that (somewhat surprisingly) theunivariatelocal limit theorem for N(H_d(n, p)) can be converted into a bivariate local limit theorem for N(H_d(n, m)) and M(H_d(n, m)). To this end, we observe that the local limit theorem for N(H_d(n, p)) implies a bivariate local limit theorem for the joint distribution of N(H_d(n, p)) and the number ¯M(H_d(n, p)) of edgesoutsidethe largest component. Then, we will set up a rela-tionship between the joint distribution ofN,M¯(Hd(n, p)) and that ofN,M¯(Hd(n, m)).

Since we already know the distribution ofN,M¯(H_d(n, p)), we can infer the joint distribu-tion ofN,M¯(H_d(n, m)) via Fourier analysis. As inH_d(n, m) thetotal number of edges is fixed (namely,m), we have ¯M(Hd(n, m)) =m− M(Hd(n, m)). Hence, we obtain a local limit theorem for the joint distribution of N,M(H_d(n, m)), i.e., Theorem 4.3. Finally, Theorem 4.3 easily implies Theorem 4.1. We actually consider this Fourier analytic ap-proach for proving the bivariate local limit theorems the main contribution of the present work.

Furthermore, in Section 5.4 we derive Theorem 5.1 from Theorem 4.1. The basic reason why this is possible is thatgiven that the largest component ofH_d(n, p) has order ν and size µ, this component is a uniformly distributed random hypergraph with these parameters. Indeed, this observation was also exploited by Łuczak [1990] to estimate the number of connected graphs up to a polynomial factor, and in Coja-Oghlan et al. [2006], where an explicit relation between Cd(ν, µ) and P[N(Hd(n, p)) =ν∧ M(Hd(n, p)) =µ]

was derived. Combining this formula with Theorem 4.1, we obtain Theorem 5.1. More-over, in Sections 5.3 and 5.5 we use similar arguments to establish Theorems 5.2 and 5.3.

The main part is organised as follows. After making some preliminaries in Section 1.3, we prove the central limit theorem forN(Hd(n, p)) via Stein’s method in Chapter 2. We outline the proof of the Local Limit Theorem 3.1 in Section 3.2. In that section we explain in detail howH_d(n, p) is generated in two “portions”. Then, in Section 3.3 we analyse the random variableS, assuming the central limit theorem forS. Further, Section 3.4 deals with the proof the central limit theorem forS via Stein’s method reusing the arguments of Chapter 2. Chapter 4 contains the proofs of additional local limit theorems for the different random graph models and joint distributions while in Chapter 5 we apply our results to get some statements about the connectivity probability and the number of connected hypergraphs.