This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self-interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal-to-noise model of soft spins in an anisotropic well, for which we prove a negative-second-moment threshold distinguishing positive from zero complexity. A universal near-critical behavior appears within this phase portrait, namely quadratic near-critical vanishing of the complexity of total critical points, and cubic near-critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non-invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.

中文翻译：

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超越不变性和弹性流形的景观复杂性

本文描述了弹性流形的退火拓扑复杂性（总临界点和局部最小值）。这种无序弹性系统的经典模型捕捉随机介质中具有自相互作用的点配置。我们在模型参数中建立了简单相图与玻璃相图，这些相由称为拉金质量的物理边界分隔开，证实了费奥多罗夫和勒杜萨尔的公式。证明的一个重要的动态步骤也适用于各向异性井中软自旋的一般信噪比模型，为此我们证明了区分正复杂度和零复杂度的负二阶矩阈值。该相图中出现了普遍的近临界行为，即总临界点复杂性的二次近临界消失和局部最小值复杂性的三次近临界消失。这两个模型充当高斯景观复杂性计算的范例，表现出很少的分布对称性，即超出不变设置。证明的两个主要输入是我们的姊妹篇论文（Ben Arous、Bourgade、McKenna 2022）中的非不变随机矩阵的行列式渐近，以及极限变分问题的非典型凸性和可积性。