Hessian spectrum at the global minimum and topology trivialization of locally isotropic Gaussian random fields
Hao Xu, Qiang Zeng
TL;DR
This work rigorously analyzes the Hessian spectrum at the global minimum for a high‑dimensional particle in a random potential with trivial topology, modeled by $H_N(x)=X_N(x)+\frac{\mu}{2}\|x\|^2$ where $X_N$ is a locally isotropic Gaussian field. It proves existence and uniqueness of the global minimum and derives the large‑$N$ limiting Hessian spectrum for both short-range (SRC) and long-range (LRC) correlation fields, confirming replica-symmetric predictions: SRC yields a semicircular spectrum with left edge at $\lambda_-^{\rm SRC}= (\sqrt{\mu}-\sqrt{B''(0)/\mu})^2$ (adjusted by scaling in the paper), while LRC yields a deformed semicircle with center $c_l=\mu+\frac{-2D''(0)}{\mu}$ and radius $r_l=\sqrt{-8D''(0)}$, with edge $c_l-r_l$. The analysis combines Kac–Rice landscape complexity, large deviation principles for GOE spectra, and matrix perturbation techniques, and extends to elastic manifolds where the limiting Hessian spectrum is described by a free additive convolution (Pastur equation). The results provide rigorous validation of replica-symmetric predictions in these Gaussian random landscapes and illuminate topology trivialization regimes, though full characterization beyond RS, particularly for elastic manifolds, remains challenging due to technical constraints on smallest-eigenvalue large deviations.
Abstract
We study the energy landscape near the ground state of a model of a single particle in a random potential with trivial topology. More precisely, we find the large dimensional limit of the Hessian spectrum at the global minimum of the Hamiltonian $X_N(x) +\frac\mu2 \|x\|^2, x\in\mathbb{R}^N,$ when $μ$ is above the phase transition threshold so that the system has only one critical point. Here $X_N$ is a locally isotropic Gaussian random field. The same idea is also applied to study the more general model of elastic manifold. In the replica symmetric regime, our results confirm the predictions of Fyodorov and Le Doussal made in 2018 and 2020 using the replica method.