We consider the imitative monomer-dimer model on the complete graph introduced in the work of Alberici et al. [J. Math. Phys. 55, 063301-1-063301-27 (2014)]. It was shown that this model is described by the monomer density and has a phase transition along certain coexistence curve, where the monomer and dimer phases coexist. More recently, it was understood [D. Alberici et al., Commun. Math. Phys. (published online, 2016)] that the monomer density exhibits the central limit theorem away from the coexistence curve and enjoys a non-normal limit theorem at criticality with normalized exponent 3/4. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein's method of exchangeable pairs. Our approach recovers what were established in the work of Alberici et al. [Commun. Math. Phys. (published online, 2016)] and furthermore allows to obtain the conditional central limit theorems along the coexistence curve. In all these results, the Berry-Esseen inequalities for the Kolmogorov-Smirnov distance are given.
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The author thanks Pierluigi Contucci for several enlightening discussions on the monomerdimer model and bringing the results in Ref. 1 to his attention, which led to the current work. The author is indebted to Qi-Man Shao for the fruitful discussions about the Stein's method and to the Department of Statistics in the Chinese University of Hong Kong for the hospitality during his visit. This research is supported by NSF Grant No. DMS-1642207, AMS-Simons Travel Grant, and Hong Kong Research Grants Council No. GRF-14302515.