Volume 9 (2026)
Aru, Juhan ; Korzhenkova, Aleksandra
The Spherical model and the large-$N$ limit of the Spin $O(N)$ model via the Gaussian free field
Annales Henri Lebesgue, Volume 9 (2026), pp. 999-1063

Metadata

DOI10.5802/ahl.281
Keywords spherical model ,  Spin O(N) model ,  Gaussian free field ,  zero-average Green’s function

Abstract

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 \rightarrow \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.


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