Results 1 
5 of
5
The order of the giant component of random hypergraphs
, 2007
"... We establish central and local limit theorems for the number of vertices in the largest component of a random duniform hypergraph Hd(n, p) with edge probability p = c / () n−1, where d−1 (d − 1) −1 + ε < c < ∞. The proof relies on a new, purely probabilistic approach, and is based on Stein’ ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
We establish central and local limit theorems for the number of vertices in the largest component of a random duniform hypergraph Hd(n, p) with edge probability p = c / () n−1, where d−1 (d − 1) −1 + ε < c < ∞. The proof relies on a new, purely probabilistic approach, and is based on Stein’s method as well as exposing the edges of Hd(n, p) in several rounds.
Local limit theorems and the number of connected hypergraphs
, 2007
"... Abstract. Let Hd(n, p) signify a random duniform hypergraph with n vertices in which each of the () n possible edges is present with probability p = p(n) independently, and let Hd(n, m) denote d a uniformly distributed with n vertices and m edges. We derive local limit theorems for the joint distri ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Let Hd(n, p) signify a random duniform hypergraph with n vertices in which each of the () n possible edges is present with probability p = p(n) independently, and let Hd(n, m) denote d a uniformly distributed with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of Hd(n, p) and Hd(n, m) for the regime () n−1 −1 p, dm/n> (d −1) +ε. As an application, we obtain an asymptotic d−1 formula for the probability that Hd(n, p) or Hd(n, m) is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in Hd(n, p) given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach. Key words: random discrete structures, giant component, local limit theorems, connected hypergraphs. 1
BROWNIAN APPROXIMATION TO COUNTING GRAPHS SOUMIK PAL
"... Abstract. Let C(n, k) denote the number of connected graphs with n labeled vertices and n+ k − 1 edges. For any sequence (kn), the limit of C(n, kn) as n tends to infinity is known. It has been observed that, if kn = o( n), this limit is asymptotically equal to the knth moment of the area under the ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Let C(n, k) denote the number of connected graphs with n labeled vertices and n+ k − 1 edges. For any sequence (kn), the limit of C(n, kn) as n tends to infinity is known. It has been observed that, if kn = o( n), this limit is asymptotically equal to the knth moment of the area under the standard Brownian excursion. These moments have been computed in the literature via independent methods. In this article we show why this is true for kn = o ( 3 n) starting from an observation made by Joel Spencer. The elementary argument uses a result about strong embedding of the Uniform empirical process in the Brownian bridge proved by Komlós, Major, and Tusnády. 1.