Fluctuations and correlations in hexagonal optical patterns

We analyze the influence of noise in transverse hexagonal patterns in nonlinear Kerr cavities. The near-field fluctuations are determined by the neutrally stable Goldstone modes associated to translational invariance and by the weakly damped soft modes. However, these modes do not contribute to the far-field intensity fluctuations that are dominated by damped perturbations with the same wave vectors than the pattern. We find strong correlations between the intensity fluctuations of any arbitrary pair of wave vectors of the pattern. Correlation between pairs forming $120°$ is larger than between pairs forming $180°,$ contrary to what a naive interpretation of emission in terms of twin photons would suggest.