Conditional Gaussian Fluctuations and Refined Asymptotics of the Spin in the Phase-Coexistence Region

We present four results on the fluctuations of the spin per site around the thermodynamic magnetization in the mean-field Blume-Capel model. Our first two results refine the main theorem in a previous paper (Ellis et al. in Ann. Appl. Probab. 20:2118–2161, ), in which the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model is given. Our first main result studies the asymptotics of the centered, finite-size magnetization, giving its precise rate of convergence to 0 along parameter sequences lying in the phase-coexistence region and converging sufficiently slowly to either a second-order point or the tricritical point of the model. A simple inequality yields our second main result, which generalizes the main theorem in Ellis et al. (Ann. Appl. Probab. 20:2118–2161, ) by giving an upper bound on the rate of convergence to 0 of the absolute value of the difference between the finite-size magnetization and the thermodynamic magnetization. These first two results have direct relevance to the theory of finite-size scaling. They are consequences of our third main result. This is a new conditional limit theorem for the spin per site, where the conditioning allows us to focus on a neighborhood of the pure states having positive thermodynamic magnetization. Our fourth main result is a conditional central limit theorem showing that the fluctuations of the spin per site are Gaussian in a neighborhood of the pure states having positive thermodynamic magnetization.