Uniqueness of Parisi measures for enriched convex vector spin glass
Hong-Bin Chen, Victor Issa
TL;DR
This work extends the uniqueness of Parisi measures from scalar to vector-spin mean-field spin glasses by studying the enriched free energy $f(t,q)$, where the enrichment is encoded by a strictly increasing path $q$ in $S^D_+$. The authors establish a modified Hopf–Lax representation and prove Gateaux and Fréchet differentiability of $f$, enabling an envelope-theorem argument that yields a unique Parisi measure $p = \nabla_q f(t,q)$ at each $(t,q)$ with $t>0$ and $q \in \mathcal{Q}_{\infty,\uparrow}$. The results show that the limit free energy satisfies a Hamilton–Jacobi type PDE and that the unique optimizer can be read off from the gradient of $f$, both in the standard and critical-point representations. These findings generalize scalar-spin uniqueness results to vector spins under strict convexity and superlinearity of the interaction $\xi$, providing a robust framework for understanding the enriched Parisi structure and overlap distribution in high-dimensional spin glasses. The work also clarifies when and how the enriched model captures the full Parisi landscape and offers tools (Hopf–Lax formula, convex duality) potentially useful for nonconvex extensions.
Abstract
In the PDE approach to mean-field spin glasses, it has been observed that the free energy of convex spin glass models could be enriched by adding an extra parameter in its definition, and that the thermodynamic limit of the enriched free energy satisfies a partial differential equation. This parameter can be thought of as a matrix-valued path, and the usual free energy is recovered by setting this parameter to be the constant path taking only the value $0$. Furthermore, the enriched free energy can be expressed using a variational formula, which is a natural extension of the Parisi formula for the usual free energy. For models with scalar spins the Parisi formula can be expressed as an optimization problem over a convex set, and it was shown in [arXiv:1402.5132] that this problem has a unique optimizer thanks to a strict convexity property. For models with vector spins, the Parisi formula cannot easily be written as a convex optimization problem. In this paper, we generalize the uniqueness of Parisi measures proven in [arXiv:1402.5132] to the enriched free energy of models with vector spins when the extra parameter is a strictly increasing path. Our approach relies on a Gateaux differentiability property of the free energy and the envelope theorem.