AI 中文总结
该论文针对球形p自旋哈密顿量，分离了正则化体普适性与无约束退火复杂度，通过矩匹配建立正则化压力普适性条件，证明高斯二次能量偏移上界并给出精确普适性的条件约化。
AI 中文摘要
固定p≥3，我们针对具有独立非高斯张量坐标的纯球形p自旋哈密顿量，建立了正则化体普适性与无约束退火复杂度之间的分离关系。存在一个对称、紧支集且具有光滑密度的无序律，其匹配前2p个高斯矩；对于该无序律，一点Kac-Rice泛函的正框架平均正则化项的期望渐近于其高斯对应项，而紧能量窗口内的无约束临界点计数具有严格高于高斯极限的更低指数速率。这种阻碍源于一个包含约N^(1/p)个坐标的相干块，因此有限矩匹配或均匀项界均无法恢复未正则化退火普适性。对于匹配前m个高斯矩的一致亚指数无序，非相干框架上的正则化泛函满足|log(E Z_{N,L}^J / E Z_{N,L}^G)| ≤ C N q_N^{m-1}，其中q_N = L^{p-1} N^{-(p-2)/2} (log N)^{(p-1)/2}。因此，均值与方差匹配意味着对所有p≥3均成立正则化压力普适性，且当(m-1)(p-2)>2时期望比趋于1。我们确定了高斯变分极限，并在高斯速率轮廓尾条件下，对渐近支撑于o(N)个坐标的轮廓一致证明了高斯二次能量偏移上界。最后，我们给出了精确普适性的条件约化：一旦两个单侧去正则化缺陷消失，精确max公式将使局部补集的控制成为必要且充分条件；该缺陷在高斯模型中消失。
英文摘要
Fix $p\ge 3$. We establish a separation between regularized bulk universality and unrestricted annealed complexity for the pure spherical $p$-spin Hamiltonian with independent non-Gaussian tensor coordinates. There is a symmetric, compactly supported disorder law with a smooth density, matching the first $2p$ Gaussian moments, for which the expectation of a positive, frame-averaged regularization of the one-point Kac-Rice functional is asymptotic to its Gaussian counterpart, while the unrestricted critical-point count in a compact energy window has a lower exponential rate strictly above the Gaussian limit. The obstruction is a coherent block involving on the order of $N^{1/p}$ coordinates, so neither finite moment matching nor a uniform entry bound restores unregularized annealed universality. For uniformly subexponential disorder matching the first $m$ Gaussian moments, the regularized functional on incoherent frames satisfies $\left|\log\left(\mathbb{E}\mathcal{Z}_{N,L}^J/\mathbb{E}\mathcal{Z}_{N,L}^G\right)\right|\le C N q_N^{m-1}$, where $q_N=L^{p-1}N^{-(p-2)/2}(\log N)^{(p-1)/2}$. Thus mean and variance matching imply regularized pressure universality for every $p\ge 3$, and the expectation ratio tends to one when $(m-1)(p-2)>2$. We identify the Gaussian variational limit and prove the Gaussian quadratic energy-excursion upper bound uniformly over profiles asymptotically supported on $o(N)$ coordinates under a Gaussian-rate profile-tail condition. Finally, we give a conditional reduction to exact universality: once two one-sided de-regularization defects vanish, an exact max formula makes control of the localized complement necessary and sufficient. The defects vanish in the Gaussian model.
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