  We study two one-parameter families of point processes connected to random
matrices: the Sine_beta and Sch_tau processes. The first one is the bulk point
process limit for the Gaussian beta-ensemble. For beta=1, 2 and 4 it gives the
limit of the GOE, GUE and GSE models of random matrix theory. In particular,
for beta=2 it is a determinantal point process conjectured to have similar
behavior to the critical zeros of the Riemann zeta-function. The second process
can be obtained as the bulk scaling limit of the spectrum of certain discrete
one-dimensional random Schr\"odinger operators.
  Both processes have asymptotically constant average density, in our
normalization one expects close to lambda/(2pi) points in a large interval of
length lambda. Our main results are large deviation principles for the average
densities of the processes, essentially we compute the asymptotic probability
of seeing an unusual average density in a large interval. Our approach is based
on the representation of the counting functions of these processes using
stochastic differential equations. We also prove path level large deviation
principles for the arising diffusions.
  Our techniques work for the full range of parameter values. The results are
novel even in the classical beta=1, 2 and 4 cases for the Sine_beta process.
They are consistent with the existing rigorous results on large gap
probabilities and confirm the physical predictions made using log-gas
arguments.
