  We consider self-loops and multiple edges in the configuration model as the
size of the graph tends to infinity. The interest in these random variables is
due to the fact that the configuration model, conditioned on being simple, is a
uniform random graph with prescribed degrees. Simplicity corresponds to the
absence of self-loops and multiple edges. We show that the number of self-loops
and multiple edges converges in distribution to two independent Poisson random
variables when the second moment of the empirical degree distribution
converges. We also provide an estimate on the total variation distance between
the number of self-loops and multiple edges and the Poisson limit of their sum.
This revisits previous works of Bollob\'as, of Janson, of Wormald and others.
The error estimates also imply sharp asymptotics for the number of simple
graphs with prescribed degrees. The error estimates follow from an application
of Stein's method for Poisson convergence, which is a novel method for this
problem. The asymptotic independence of self-loops and multiple edges follows
from a Poisson version of the Cram\'er-Wold device using thinning, which is of
independent interest. When the degree distribution has infinite second moment,
our general results break down. We can, however, prove a central limit theorem
for the number of self-loops, and for the multiple edges between vertices of
degrees much smaller than the square root of the size of the graph, or when we
truncate the degrees similarly. Our results and proofs easily extend to
directed and bipartite configuration models.
