  In the random graph $G(n,p)$ with $pn$ bounded, the degrees of the vertices
are almost i.i.d Poisson random variables with mean $\gl:= p(n-1)$. Motivated
by this fact, we introduce the Poisson cloning model $G_{PC} (n,p)$ for random
graphs in which the degrees are i.i.d Poisson random variables with mean $\gl$.
Then, we first establish a theorem that shows the new model is equivalent to
the classical model $G(n,p)$ in an asymptotic sense. Next, we introduce a
useful algorithm, called the cut-off line algorithm, to generate the random
graph $G_{PC} (n,p)$. The Poisson cloning model $G_{PC}(n,p)$ equipped with the
cut-off line algorithm enables us to very precisely analyze the sizes of the
largest component and the $t$-core of $G(n,p)$. This new approach to the
problems yields not only elegant proofs but also improved bounds that are
essentially best possible. We also consider the Poisson cloning models for
random hypergraphs and random $k$-SAT problems. Then, the $t$-core problem for
random hypergraphs and the pure literal algorithm for random $k$-SAT problems
are analyzed.
