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Limit theorems in the imitative monomer-dimer mean-field model via Stein’s method
Alberici, D. , Contucci, P. , Fedele, M. , and Mingione, E. , “Limit theorems for monomer-dimer mean-field models with attractive potential,” Commun. Math. Phys. (published online, 2016).
Alberici, D. , Contucci, P. , and Mingione, E. , “A mean-field monomer-dimer model with attractive interaction. Exact solution and rigorous results,” J. Math. Phys. 55, 063301-1–063301-27 (2014).
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Eichelsbacher, P. and Löwe, M. , “Stein’s method for dependent random variables occurring in statistical mechanics,” Electron. J. Probab. 15(30), 962–988 (2010).
Ellis, R. S. and Newman, C. M. , “Limit theorems for sums of dependent random variables occurring in statistical mechanics,” Probab. Theory Relat. Fields 44, 117-139 (1978).
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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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