  A one-parameter family of point processes describing the distribution of the
critical points of the characteristic polynomial of large random Hermitian
matrices on the scale of mean spacing is investigated. Conditionally on the
Riemann hypothesis and the multiple correlation conjecture, we show that one of
these limiting processes also describes the distribution of the critical points
of the Riemann $\xi$-function on the critical line.
  We prove that each of these processes boasts stronger level repulsion than
the sine process describing the limiting statistics of the eigenvalues: the
probability to find $k$ critical points in a short interval is comparable to
the probability to find $k+1$ eigenvalues there. We also prove a similar
property for the critical points and zeros of the Riemann $\xi$-function,
conditionally on the Riemann hypothesis but not on the multiple correlation
conjecture.
