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

We revisit the relation between the spherical model of Berlin-Kac and the spin $O(N)$ model in the limit $N \to \infty$, and explain how they are connected via the discrete Gaussian free field (GFF). 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 $N \to \infty$ 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. In particular, although the limiting free energies per site of the two models agree at all temperatures, their finite-dimensional laws still differ in terms of their zero modes in the low-temperature regime.