We quantify the concept of ensembles of atoms' fluctuations for 2D liquids close to and in the two-phase coexistence region, using a probabilistic method of local structure analysis. Two ensembles are studied: gas-like independent gaussian fluctuations (IGF) A.C. Mituœ, A.Z. Patashinski, A. Patrykiejew, S. Sokolowski,  Phys. Rev. B66, 184202 (2002) and a truly solid-like fluctuations (SF) characteristic for a two-dimensional (2D) triangular solid. For a 2D Lennard--Jones (LJ) system simulated using Monte Carlo and molecular dynamics methods those ensembles yield a statistically reliable description of local solid-like structures in the two-phase region. Both ensembles undergo a spectacular breakdown as the density changes. A hypothetical relation of this breakdown to the behaviour of the heat capacity in the two-phase region is proposed.

PACS numbers: 61.20.Ja, 64.70.Dv