The $\ell^p$-Gaussian-Grothendieck problem with vector spins

We study the vector spin generalization of the $\ell^p$-Gaussian-Grothendieck problem. In other words, given integer $\kappa\geq 1$, we investigate the asymptotic behaviour of the ground state energy associated with the Sherrington-Kirkpatrick Hamiltonian indexed by vector spin configurations in the unit $\ell^p$-ball. The ranges $1\leq p\leq 2$ and $2<p<\infty$ exhibit significantly different behaviours. When $1\leq p\leq 2$, the vector spin generalization of the $\ell^p$-Gaussian-Grothendieck problem agrees with its scalar counterpart. In particular, its re-scaled limit is proportional to some norm of a standard Gaussian random variable. On the other hand, for $2<p<\infty$ the re-scaled limit of the $\ell^p$-Gaussian-Grothendieck problem with vector spins is given by a Parisi-type variational formula.