Our goal is to discuss in detail the calculation of the mean number of
stationary points and minima for random isotropic Gaussian fields on a sphere
as well as for stationary Gaussian random fields in a background parabolic
confinement. After developing the general formalism based on the
high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory
(RMT) techniques to perform analysis of the random energy landscape of $p-$spin
spherical spinglasses and a related glass model, both displaying a
zero-temperature one-step replica symmetry breaking glass transition as a
function of control parameters (e.g. a magnetic field or curvature of the
confining potential). A particular emphasis of the presented analysis is on
understanding in detail the picture of "topology trivialization" (in the sense
of drastic reduction of the number of stationary points) of the landscape which
takes place in the vicinity of the zero-temperature glass transition in both
models. We will reveal the important role of the GOE "edge scaling" spectral
region and the Tracy-Widom distribution of the maximal eigenvalue of GOE
matrices for providing an accurate quantitative description of the universal
features of the topology trivialization scenario.

本研究通过基于高维 Kac-Rice 公式的一般公式及其与随机矩阵理论技术的结合，分析了零温度单步复制对称破缺玻璃转变的能量景观的随机性，提取了高斯场的平均稳定点和极小值以及一个弧形限制中的稳定高斯随机场的数量，并详细阐述了在两个模型中，玻璃转变的零温度附近出现 “拓扑平凡化” 的现象，以及 GOE “边缘缩放” 谱区和 GOE 矩阵的最大特征值的 Tracy-Widom 分布对于提供 “拓扑平凡化” 场景的普适特征的精确量化描述的重要作用。