The properties of the absorbing states of non-equilibrium models belonging to the conserved directed percolation universality class are studied. We find that at the critical point the absorbing states are hyperuniform, exhibiting anomalously small de
nsity fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.
A simple periodically driven system displaying rich behavior is introduced and studied. The system self-organizes into a mosaic of static ordered regions with three possible patterns, which are threaded by one-dimensional paths on which a small numbe
r of mobile particles travel. These trajectories are self-avoiding and non-intersecting, and their relationship to self-avoiding random walks is explored. Near $rho=0.5$ the distribution of path lengths becomes power-law like up to some cutoff length, suggesting a possible critical state.
A solid is typically deemed amorphous when there are no Bragg peaks in its diffraction pattern. We discuss a two dimensional configuration of Ising spins with an autocorrelation function which vanishes at all nonzero distances, so that its scattering
pattern is flat. This configuration is a ground state of a Hamiltonian with deterministic, translationally-invariant and finite range interactions. Despite ostensibly being amorphous, this configuration has perfect underlying order. The finite temperature behavior of this model exhibits ordering transitions at successively larger length scales.
