Hessian spectrum at the global minimum and topology trivialization of locally isotropic Gaussian random fields

2022/10/27 by Hao Xu, Qiang Zeng, Xu, Hao +1
Computer Science · #Computational Physics and Python Applications #FOS: Mathematics #FOS: Physical sciences #Image Processing and 3D Reconstruction #Mathematical Physics (math-ph) #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2210.15254

openalex publication_date 2022/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) +\fracμ2 ‖x‖², x∈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.

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