We study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of $N$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order $N/\log N$, in contrast to the $O(N)$ bound obtained from the Gaussian–Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function $f$ that arises in the representation of the disorder as $f(g)$ for $g$ standard normal. Under an additional regularity assumption on $f$, we further show that the variance is of order at most $N/\log N$.

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谢林顿-柯克帕特里克模型中超浓缩的普遍性

我们研究了谢林顿-柯克帕特里克模型中自由能超浓缩的普遍性。在[10]中，Chatterjee 表明，当系统包含$氮$自旋和高斯无序，通过建立阶数上限来超集中该量的方差$氮/\mathrm{日志}氮$，与$氧（氮）$从高斯-庞加莱不等式获得的界限。在本文中，我们证明超集中确实适用于具有有限三阶矩的中心无序的任何选择，其中上限以辅助非递减函数表示$F$出现在疾病的表征中$F（G）$为了$G$标准正常。在额外的规律性假设下$F$，我们进一步证明方差至多是有序的$氮/\mathrm{日志}氮$。