Free energy in the mixed p-spin models with vector spins

TL;DR

This paper extends the Parisi formula to mixed p-spin models with vector spins, providing a rigorous characterization of free energy and confirming Talagrand's upper bound for coupled systems.

Contribution

It introduces a Parisi formula analogue for vector spin models and proves the sharpness of Talagrand's upper bound in this context.

Findings

Derived the Parisi formula for vector spin models.

Proved the sharpness of Talagrand's upper bound.

Extended synchronization techniques to mixed p-spin systems.

Abstract

Using the synchronization mechanism developed in the previous work on the Potts spin glass model, arXiv:1512.00370, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the Sherrington-Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand's upper bound for the free energy of multiple mixed $p$-spin systems coupled by constraining their overlaps.