  Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in
which each of the ${n}\choose{d}$ possible edges is present with probability
$p=p(n)$ independently, and let $H_d(n,m)$ denote 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 $H_d(n,p)$ and $H_d(n,m)$ for the regime ${{n-1}\choose{d-1}}
p,dm/n >(d-1)^{-1}+\epsilon$. As an application, we obtain an asymptotic
formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected. In
addition, we infer a local limit theorem for the conditional distribution of
the number of edges in $H_d(n,p)$ given connectivity. While most prior work on
this subject relies on techniques from enumerative combinatorics, we present a
new, purely probabilistic approach.
