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

TL;DR

This paper explores the connection between the spherical model and the spin O(N) model as N approaches infinity, demonstrating their relation via the Gaussian free field and analyzing their behavior across temperature regimes.

Contribution

It provides a detailed probabilistic analysis of the large-N limit of the spin O(N) model and its relation to the spherical model through the Gaussian free field, including new results at the critical temperature.

Findings

Infinite-volume limit of the spherical model is a massive GFF at high temperature.

At the critical temperature, the limit is a standard GFF.

In the low-temperature regime, the limit includes a GFF plus a Rademacher or Gaussian constant.

Abstract

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.