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

TL;DR

This paper investigates the asymptotic behavior of the ground state energy in a vector spin generalization of the $\, ext{ extlbrack} p ext{ extbrack}$-Gaussian-Grothendieck problem, revealing different regimes for $p$ values and deriving a variational formula for the case $p>2$.

Contribution

It introduces a vector spin extension of the $\, extlbrack} p extbrack$-Gaussian-Grothendieck problem and characterizes its asymptotic limits, including a new variational formula for $p>2$.

Findings

For $1 \, extleq p \, extleq 2$, the problem's limit matches the scalar case and relates to Gaussian norms.

For $p>2$, the limit is described by a Parisi-type variational formula.

Different behaviors are observed in the regimes $1 \, extleq p \, extleq 2$ and $p>2$.

Abstract

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.