  We define a "renormalized energy" as an explicit functional on arbitrary
point configurations of constant average density in the plane and on the real
line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is
obtained by subtracting two leading terms from the Coulomb potential on a
growing number of charges. The functional is expected to be a good measure of
disorder of a configuration of points. We give certain formulas for its
expectation for general stationary random point processes. For the random
matrix $\beta$-sine processes on the real line (beta=1,2,4), and Ginibre point
process and zeros of Gaussian analytic functions process in the plane, we
compute the expectation explicitly. Moreover, we prove that for these processes
the variance of the renormalized energy vanishes, which shows concentration
near the expected value. We also prove that the beta=2 sine process minimizes
the renormalized energy in the class of determinantal point processes with
translation invariant correlation kernels.
