论文标题
给定径向衍生物的本地最大值的复杂性，用于混合$ p $ -spin hamiltonians
Complexity of local maxima of given radial derivative for mixed $p$-spin Hamiltonians
论文作者
论文摘要
我们研究了带有球形混合$ p $ spin模型的给定径向衍生物的局部最大值的数量，并证明第二刻的时间与任意混合物和任何径向衍生物的指数尺度上的第一矩相匹配。这是令人惊讶的，因为对于具有给定的径向衍生物和给定能量的局部最大值的数量，相应的结果仅对特定混合物[sub17; BSZ20]。我们使用标准的KAC-RICE计算来得出指数尺度上的第一和第二刻的公式，然后找到一个显着的分析参数，该论点表明，在此一般环境中，第二刻度公式的界限是第一个时刻公式的两倍。这也导致了一个新的证明中央不平等的证明，用于证明[sub17]中给定能量的纯$ p $ - 速spin模型的数字临界点的浓度，并消除了该论文中使用的计算机协助参数的需求，价格为$ 3 \ leq p \ leq leq 10 $。
We study the number of local maxima with given radial derivative of spherical mixed $p$-spin models and prove that the second moment matches the square of the first moment on exponential scale for arbitrary mixtures and any radial derivative. This is surprising, since for the number of local maxima with given radial derivative and given energy the corresponding result is only true for specific mixtures [Sub17; BSZ20]. We use standard Kac-Rice computations to derive formulas for the first and second moment at exponential scale, and then find a remarkable analytic argument that shows that the second moment formula is bounded by twice the first moment formula in this general setting. This also leads to a new proof of a central inequality used to prove concentration of the number critical points of pure $p$-spin models of given energy in [Sub17] and removes the need for the computer assisted argument used in that paper for $3 \leq p \leq 10$.
