  We investigate the dynamics of a strongly interacting spin system that is
motivated by current experimental realizations of strongly interacting Rydberg
gases in lattices. In particular we are interested in the temporal evolution of
quantities such as the density of Rydberg atoms and density-density
correlations when the system is initialized in a fully polarized state without
Rydberg excitations. We show that in the thermodynamic limit the expectation
values of these observables converge at least logarithmically to universal
functions and outline a method to obtain these functions. We prove that a
finite one-dimensional system follows this universal behavior up to a given
time. The length of this universal time period depends on the actual system
size. This shows that already the study of small systems allows to make precise
predictions about the thermodynamic limit provided that the observation time is
sufficiently short. We discuss this for various observables and for systems
with different dimensions, interaction ranges and boundary conditions.
