论文标题
复杂的复杂景观
Complex complex landscapes
论文作者
论文摘要
我们研究$ p $ -spin模型的马鞍点 - 当它的$ n $变量很复杂时，最好理解的“复合体”（崎ged）景观的示例。这些点是$ n $随机方程$ p-1 $的系统的解决方案。我们解决$ \ overline {\ Mathcal n} $，在$ n \ to \ infty $限制中平均的解决方案数量。我们发现它使BézoutBound $ \ log \ overline {\ Mathcal n} \ sim n \ log（p-1）$饱和。每个鞍座的黑森斯由$ c^\匕首c $的随机矩阵给出，其中$ c $是一个复杂的对称高斯矩阵，转移到对角线。它的频谱具有一个过渡，差距会发展出概括在实际问题中众所周知的“阈值级别”的概念。实际问题的结果以实际参数的限制恢复。在这种情况下，仅解决方案总数的平方根是真实的。就复杂能量而言，解决方案被分为马鞍具有不同拓扑特性的部门。
We study the saddle-points of the $p$-spin model -- the best understood example of a `complex' (rugged) landscape -- when its $N$ variables are complex. These points are the solutions to a system of $N$ random equations of degree $p-1$. We solve for $\overline{\mathcal N}$, the number of solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the Bézout bound $\log\overline{\mathcal N}\sim N\log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger C$, where $C$ is a complex symmetric Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the square-root of the total number of solutions are real. In terms of the complex energy, the solutions are divided into sectors where the saddles have different topological properties.
