  For a broad class of point processes, including determinantal point
processes, we construct associated marked and conditional ensembles, which
allow to study a random configuration in the point process, based on
information about a randomly incomplete part of the configuration. We show that
our construction yields a well behaving transformation of sufficiently regular
point processes. In the case of determinantal point processes, we explain that
special cases of the conditional ensembles already appear implicitly in the
literature, namely in the study of unitary invariant random matrix ensembles,
in the Its-Izergin-Korepin-Slavnov method to analyze Fredholm determinants, and
in the study of number rigidity. As applications of our construction, we show
that a class of determinantal point processes induced by orthogonal projection
operators, including the sine, Airy, and Bessel point processes, satisfy a
strengthened notion of number rigidity, and we give a probabilistic
interpretation of the Its-Izergin-Korepin-Slavnov method.
