  We consider random permutation matrices following a one-parameter family of
deformations of the uniform distribution, called Ewens' measures, and
modifications of these matrices where the entries equal to one are replaced by
i.i.d uniform random variables on the unit circle. For each of these two
ensembles of matrices, rescaling properly the eigenangles provides a limiting
point process as the size of the matrices goes to infinity. If $J$ is an
interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity,
the number of points lying in $J$ of the limiting point process related to
modified permutation matrices is asymptotically normal. Moreover, for
permutation matrices without modification, if $a$ and $a+b$ denote the
endpoints of $J$, we still have an asymptotic normality for the number of
points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$]
and [$a,b \to \infty$ with $b$ proportional to $a$].

Probability