The Spherical model and the large- N limit of the Spin O ( N ) model via the Gaussian free field

Using probabilistic limit theorems and concentration of measure, we prove that the infinite-volume limit of the spherical model on a d -dimensional torus is a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Rademacher random constant in the low-temperature regime. The proof at the critical temperature appears to be new and relies on a fine analysis of the zero-average Green’s function on the torus. We study the spin O ( N ) model in the double limit of spin dimensionality and torus size. Sending N → ∞ first, and then the torus size to infinity, we show that the different spin coordinates become i.i.d. fields, distributed as a massive GFF in the high-temperature regime, a standard GFF at the critical temperature, and a standard GFF plus a Gaussian random constant in the low-temperature regime.