We obtain explicit expressions for the annealed complexities associated, respectively, with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension $d<4$ embedded in a random medium of dimension $N\gg 1$ and confined by a parabolic potential with the curvature parameter $\mu$. These complexities are found to both vanish at the critical value ${\mu }_c$ identified as the Larkin mass. For $\mu <{\mu }_c$ the system is in complex phase corresponding to the replica symmetry breaking in its $T=0$ thermodynamics. The complexities vanish, respectively, quadratically (stationary points) and cubically (minima) at ${\mu }_c^{-}$. For $d\ge 1$ they admit a finite “massless” limit $\mu =0$ which is used to provide an upper bound for the depinning threshold under an applied force.

中文翻译：

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高维随机景观中的流形：固定点和固定的复杂性

对于具有内部尺寸的弹性歧管，我们分别获得了与（i）固定点和（ii）能量景观的局部极小值总数相关的退火复杂度的显式表达式 $d<4$ 嵌入尺寸随机介质中 $ñ\gg \mathrm{1个}$ 并由带有曲率参数的抛物线势限制 $\mu$。发现这些复杂性都在临界值上消失了${\mu }_C$确定为拉金质量。对于$\mu <{\mu }_C$ 该系统处于复杂阶段，对应于其复制品对称性破裂 $Ť=0$热力学。复杂度分别在（二次）（平稳点）和（三次）（最小）处消失${\mu }_C^{-}$。对于$d\ge \mathrm{1个}$ 他们承认有限的“无质量”限制 $\mu =0$ 它用于在施加力的作用下为脱钉阈值提供上限。