On a spherically lifted spin model at finite temperature

We investigate an $n$-vector model over $k$ sites with generic pairwise interactions and spherical constraints. The model is a lifting of the Ising model whereby the support of the spin is lifted to a hypersphere. We show that the $n$-vector model converges to a limiting distribution at a rate of $n^{-1/2 + o(1)}$. We show that the limiting distribution for $n \to \infty$ is determined by the solution of an equality-constrained maximization task over positive definite matrices. We prove that the obtained maximal value and maximizer, respectively, give rise to the free energy and correlation function of the limiting distribution. In the finite temperature regime, the maximization task is a log-determinant regularization of the semidefinite program (SDP) in the Goemans-Williamson algorithm. Moreover, the inverse temperature determines the regularization strength, with the zero temperature limit converging to the SDP in Goemans-Williamson. Our derivation draws a curious connection between the semidefinite relaxation of integer programming and the spherical lifting of sampling on a hypercube. To the authors' best knowledge, this work is the first to solve the setting of fixed $k$ and infinite $n$ under unstructured pairwise interactions.