We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature $$\mu $$ μ . We are interested in the mean spectral density $$\rho (\lambda )$$ ρ ( λ ) of the Hessian matrix $${{\mathcal {K}}}$$ K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for $$\rho (\lambda )$$ ρ ( λ ) for a fixed $$L^d$$ L d in the $$N \rightarrow \infty $$ N → ∞ limit extending $$d=0$$ d = 0 results of our previous work (Fyodorov et al. in Ann Phys 397:1–64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting $$L\rightarrow \infty $$ L → ∞ . In all cases we show that for a confinement curvature $$\mu $$ μ exceeding a critical value $$\mu _c$$ μ c , the so-called “Larkin mass”, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at $$\mu \rightarrow \mu _c$$ μ → μ c . For $$\mu <\mu _c$$ μ < μ c the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as $$(\mu _c-\mu )^4$$ ( μ c - μ ) 4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called “marginal cases” in $$d=1,2$$ d = 1 , 2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.

中文翻译：

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由高维随机景观固定的流形：全球能量最小值处的 Hessian

我们考虑一个内部维度为 d 和长度为 L 的弹性流形，其固定在 N 维随机势中，并受曲率 $\mu $ μ 的附加抛物线势的限制。我们对 Hessian 矩阵 $${{\mathcal {K}}}$K 在总能量的绝对最小值处的平均谱密度 $$\rho (\lambda )$$ ρ ( λ ) 感兴趣。我们使用复制方法推导出 $$\rho (\lambda )$$ ρ ( λ ) 的方程组，用于 $$N \rightarrow \infty $$ N 中的固定 $$L^d$$ L d → ∞ 限制扩展 $$d=0$$ d = 0 我们之前工作的结果（Fyodorov 等人，Ann Phys 397:1–64, 2018）。特别注意通过让 $$L\rightarrow \infty $$ L → ∞ 来分析扩展格系统的极限。在所有情况下，我们表明，对于超过临界值 $$\mu _c$$ μ c 的限制曲率 $$\mu $$ μ c ，所谓的“拉金质量”，系统是复制对称的，Hessian 谱总是有间隙的（从零开始）。间隙在 $$\mu \rightarrow \mu _c$$ μ → μ c 处二次消失。对于 $$\mu <\mu _c$$ μ < μ c，发生复制对称性破坏 (RSB) 并且 Hessian 谱要么有间隙，要么向下延伸到零，具体取决于 RSB 是 1 步还是满。在 1-RSB 的情况下，所有 d 中的差距都消失了，因为 $$(\mu _c-\mu )^4$$ ( μ c - μ ) 4 接近过渡。在完整的 RSB 情况下，间隙完全为零。一组特定的景观在 $$d=1,2$$ d = 1 , 2 中实现了所谓的“边缘情况”，它们共享 1-step 和完整 RSB 解决方案的特征，并表现出一定的尺度不变性。