  We establish sharp global rigidity upper bounds for universal determinantal
point processes describing edge eigenvalues of random matrices. For this, we
first obtain a general result which can be applied to general (not necessarily
determinantal) point processes which have a smallest (or largest) point: it
allows to deduce global rigidity upper bounds from the exponential moments of
the counting function of the process. By combining this with known exponential
moment asymptotics for the Airy and Bessel point processes, we improve on the
best known upper bounds for the global rigidity of the Airy point process, and
we obtain new global rigidity results for the Bessel point process.
  Secondly, we obtain exponential moment asymptotics for the Wright's
generalized Bessel process and the Meijer-$\mathrm{G}$ process, up to and
including the constant term. As a direct consequence, we obtain new results for
the expectation and variance of the associated counting functions. Furthermore,
by combining these asymptotics with our general rigidity theorem, we obtain new
global rigidity upper bounds for these point processes.
