  We consider large Information-Plus-Noise type matrices of the form
$M_N=(\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac{X_N}{\sqrt{N}}+A_N)^*$
where $X_N$ is an $n \times N$ ($n\leq N)$ matrix consisting of independent
standardized complex entries, $A_N$ is an $n \times N$ nonrandom matrix and
$\sigma>0$. As $N$ tends to infinity, if $n/N \rightarrow c\in ]0,1]$ and if
the empirical spectral measure of $A_N A_N^*$ converges weakly to some
compactly supported probability distribution $\nu \neq \delta_0$, Dozier and
Silverstein established that almost surely the empirical spectral measure of
$M_N$ converges weakly towards a nonrandom distribution $\mu_{\sigma,\nu,c}$.
Bai and Silverstein proved, under certain assumptions on the model, that for
some closed interval in $]0;+\infty[$ outside the support of
$\mu_{\sigma,\nu,c}$ satisfying some conditions involving $A_N$, almost surely,
no eigenvalues of $M_N$ will appear in this interval for all $N$ large. In this
paper, we carry on with the study of the support of the limiting spectral
measure previously investigated by Dozier and Silverstein and later by Vallet,
Loubaton and Mestre and Loubaton and P. Vallet, and we show that, under almost
the same assumptions as Bai and Silvertein, there is an exact separation
phenomenon between the spectrum of $M_N$ and the spectrum of $A_NA_N^*$: to a
gap in the spectrum of $M_N$ pointed out by Bai and Silverstein, it corresponds
a gap in the spectrum of $A_NA_N^*$ which splits the spectrum of $A_NA_N^*$
exactly as that of $M_N$. We use the previous results to characterize the
outliers of spiked Information-Plus-Noise type models.
