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Summary form only given. Transverse optical patterns are capable of allowing noteworthy aspects linked to quantum fluctuations. These issues have been mainly studied in stripe patterns, which present only two modes in the far-field. We address the study of the fluctuations and correlations in a hexagonal pattern in a ring cavity filled with a self-focusing Kerr medium. The dynamics of the electric field is described, in the mean-field approximation, by a Langevin equation for the scaled slowly varying amplitude of the field. We show that the pattern fluctuations can be understood in terms of the neutral modes of the pattern and modes with small damping. These modes are obtained from a stability analysis of the hexagonal pattern. We find strong correlations between the intensity fluctuations of the modes of the pattern, not only among the fundamental harmonics, but also with the higher order ones. We also find strong anticorrelations between the intensity fluctuations of the modes of the pattern and the homogeneous mode which are related to energy conservation.