HESSIAN SPECTRUM AT THE GLOBAL MINIMUM AND TOPOLOGY TRIVIALIZATION OF LOCALLY ISOTROPIC GAUSSIAN RANDOM FIELDS

• 成果类型: Article
• 署名作者: Xu, Hao; Zeng, Qiang
• 署名单位: University of Macau; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/25-AAP2152
• 发表日期: 2025
• 页码: 1668-1715
• 关键词: replica symmetry-breaking; large deviations; RANDOM MATRICES; complexity; models

摘要

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 XN(x) + mu 2 HxH2, x E RN, when mu is above the phase transition threshold so that the system has only one critical point. Here XN 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.