Vector spin glasses with Mattis interaction I: the convex case

This paper constitutes the first part of a two-paper series devoted to the systematic study of vector spin glass models whose energy function involves a spin glass part and a general Mattis interaction part. In this paper, we focus on models whose spin glass part satisfies the usual convexity assumption. We identify the limit free energy via a Parisi-type formula and prove a large deviation principle for the mean magnetization. The proof is remarkably simple and short compared to previous approaches; it relies on treating the Mattis interaction as a parameter of the model. In the companion paper, we establish similar results in the high-temperature regime for models whose spin glass part is not assumed to satisfy the usual convexity assumption.

💡 Research Summary

This paper is the first in a two‑part series devoted to a systematic treatment of vector spin‑glass models whose Hamiltonian consists of a conventional spin‑glass term together with a general Mattis interaction. The focus here is on models whose spin‑glass part satisfies the usual convexity assumption (the covariance function ξ is convex on the cone of positive semidefinite matrices). The authors identify the limiting free energy through a Parisi‑type variational formula and establish a quenched large‑deviation principle (LDP) for the mean magnetization.

The model is defined as follows. Spins are D‑dimensional vectors σ_i∈ℝ^D drawn i.i.d. from a probability measure P₁ supported on the unit ball and spanning ℝ^D. The spin‑glass part is a centered Gaussian field H_N(σ) with covariance E