Strong Topological Trivialization of Multi-Species Spherical Spin Glasses

TL;DR

The paper investigates landscapes of mean-field multi-species spherical spin glasses, deriving a phase boundary for annealed trivialization of critical-point counts and proving its equivalence to a quenched strong topological trivialization (STT). Employing an enhanced Kac–Rice framework coupled with a vector Dyson equation, it first identifies the annealed trivial regime and then sharpens the description to show exactly $2^r$ well-conditioned critical points in the strictly super-solvable setting, with all approximate critical points closely surrounding these true points; this culminates in logarithmic-time Langevin mixing at low temperature. In the strictly sub-solvable regime, the authors (in a companion work) construct exponentially many well-separated approximate critical points, demonstrating quenched non-trivialization and highlighting a phase transition between regimes. The results bridge landscape topology with algorithmic/dynamical behavior, offering a robust methodology to analyze non-convex random landscapes via controlled determinants and band localization, and yielding concrete consequences for sampling and optimization in complex systems.

Abstract

We study the landscapes of multi-species spherical spin glasses. Our results determine the phase boundary for annealed trivialization of the number of critical points, and establish its equivalence with a quenched strong topological trivialization property. Namely in the "trivial" regime, the number of critical points is constant, all are well-conditioned, and all approximate critical points are close to a true critical point. As a consequence, we deduce that Langevin dynamics at sufficiently low temperature has logarithmic mixing time. Our approach begins with the Kac--Rice formula. We characterize the annealed trivialization phase by explicitly solving a suitable multi-dimensional variational problem, obtained by simplifying certain asymptotic determinant formulas from (Ben Arous--Bourgade--McKenna 2023, McKenna 2024). To obtain more precise quenched results, we develop general purpose techniques to avoid sub-exponential correction factors and show non-existence of approximate critical points. Many of the results are new even in the 1-species case.

Figures (2)

• Figure 1: Figure \ref{['subfig:one-species']}: the complexity functional $F$ of a $1$-species model is shown. $F$ is tangent to the $x$-axis at two global maxima marked by green X's. The red X is a local minimum. The two dashed vertical lines mark the transition from local convexity to concavity, and $F"$ is discontinuous at these points. Figures \ref{['subfig:two-species']} and \ref{['subfig:two-species-weird']}: points of interest are shown in the domain ${\mathbb{R}}^2$ of the complexity functionals $F$ for two different $2$-species models. The green X's are global maxima where $F$ equals $0$, while the red X's are stationary points that are not local maxima. The blue boundary is analogous to the dashed vertical lines in Figure \ref{['subfig:one-species']}, and is where ${\vec{m}}(0;\vec{x})$ transitions from real and nonreal. In the four regions outside this boundary, ${\vec{m}}(0;\vec{x})$ is real, and in the region inside it ${\vec{m}}(0;\vec{x})$ is non-real. By Lemma \ref{['lem:u-manifold']} and continuity of $\vec{x} \mapsto {\vec{m}}(0;\vec{x})$ (see Theorem \ref{['thm:continuity']}), this boundary is also the set of $\vec{x}$ for which ${\vec{m}}(0;\vec{x})$ is real and $M({\vec{m}}(0;\vec{x}))$ is singular. In Figure \ref{['subfig:two-species']}, $F$ is locally non-concave inside this boundary, but in Figure \ref{['subfig:two-species-weird']}$F$ is also locally concave in the shaded purple regions. Note also that in Figure \ref{['subfig:two-species-weird']}, there are only three red X's instead of five; the $3^r$ stationary points identified by Lemma \ref{['lem:stationary-condition']} do not necessarily all exist.
• Figure 2: A diagram of the ellipsoid $S$, used in the proof of Proposition \ref{['prop:kac-rice-E-infinity']}. If the tangent line to $S$ at $(E_{\infty}^+,\sqrt 2)$ has positive slope, then the red region is empty and we conclude \ref{['eq:kac-rice-E-infinity-plus']}. If not, all local maxima correspond to the blue region, hence have energy at most $E_{\infty}^+ + o_N(1)$. This implies that $GS(\xi)\leq E_{\infty}^+$.

Theorems & Definitions (201)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Definition 5
• Theorem 1.5
• Definition 6