The ℓp-Gaussian-Grothendieck problem with vector spins

Abstract

We study the vector spin generalization of the ℓp-Gaussian-Grothendieck problem. In other words, given integer κ≥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 ℓp-ball. The ranges 1≤p≤2 and 2<p<∞ exhibit significantly different behaviours. When 1≤p≤2, the vector spin generalization of the ℓp-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<∞ the re-scaled limit of the ℓp-Gaussian-Grothendieck problem with vector spins is given by a Parisi-type variational formula.