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。

高维随机场与随机矩阵理论

High-Dimensional Random Fields and Random Matrix Theory

We reveal a phase transition with decreasing viscosity $\nu$ at \nu=\nu_c>0
in one-dimensional decaying Burgers turbulence with a power-law correlated
random profile of Gaussian-distributed initial velocities
<v(x,0)v(x',0)>\sim|x-x'|^{-2}. The low-viscosity phase exhibits non-Gaussian
one-point probability density of velocities, continuously dependent on \nu,
reflecting a spontaneous one step replica symmetry breaking (RSB) in the
associated statistical mechanics problem. We obtain the low orders cumulants
analytically. Our results, which are checked numerically, are based on
combining insights in the mechanism of the freezing transition in random
logarithmic potentials with an extension of duality relations discovered
recently in Random Matrix Theory. They are essentially non mean-field in nature
as also demonstrated by the shock size distribution computed numerically and
different from the short range correlated Kida model, itself well described by
a mean field one step RSB ansatz. We also provide some insights for the finite
viscosity behaviour of velocities in the latter model.

通过应用随机矩阵理论中最新发现的对偶关系，本研究发现一维减小的 Burgers 湍流在粘度为 $
u_c>0$ 时出现相变，表现为非高斯的自发单级别对称性破缺，这为后续研究提供了基础和指导。

瓦解的 Burgers 湍流和随机矩阵对偶中的冻结转变

Freezing Transition in Decaying Burgers Turbulence and Random Matrix  Dualities

So far the problem of a spin glass on a Bethe lattice has been solved only at
the replica symmetric level, which is wrong in the spin glass phase. Because of
some technical difficulties, attempts at deriving a replica symmetry breaking
solution have been confined to some perturbative regimes, high connectivity
lattices or temperature close to the critical temperature.
Using the cavity method, we propose a general non perturbative solution of
the Bethe lattice spin glass problem at a level of approximation which is
equivalent to a one step replica symmetry breaking solution. The results
compare well with numerical simulations. The method can be used for many finite
connectivity problems appearing in combinatorial optimization.

使用空腔方法，我们提出了一种非微扰的 Bethe 格子自旋玻璃问题的解决方法，其逼近等级等同于单步复制对称性破缺解决方案，并且可以用于出现在组合优化中的许多有限连接问题。

重访 Bethe 树自旋玻璃

The Bethe lattice spin glass revisited

We analytically derive the geometrical structure of the weight space in
multilayer neural networks (MLN), in terms of the volumes of couplings
associated to the internal representations of the training set. Focusing on the
parity and committee machines, we deduce their learning and generalization
capabilities both reinterpreting some known properties and finding new exact
results. The relationship between our approach and information theory as well
as the Mitchison--Durbin calculation is established. Our results are exact in
the limit of a large number of hidden units, showing that MLN are a class of
exactly solvable models with a simple interpretation of replica symmetry
breaking.

对于多层神经网络中的权值空间，MLN 是精确可解的模型，具有复制对称性，通过内部表达与训练集相关的偶奇和委员会机器等，推导其学习和泛化能力，并找到了一些新的精确结果。