  We define a notion of logarithmic, Coulomb and Riesz interactions in any
dimension for random systems of infinite charged point configurations with a
uniform background of opposite sign. We connect this interaction energy with
the "renormalized energy" studied by Serfaty et al., which appears in the free
energy functional governing the microscopic behavior of logarithmic, Coulomb
and Riesz gases. Minimizers of this functional include the Sine-beta processes
in the one-dimensional Log-gas case. Using our explicit expression (inspired by
the work of Borodin-Serfaty) we prove their convergence to the Poisson process
in the high-temperature limit as well as a crystallization result in the
low-temperature limit for one-dimensional systems.
