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Abstract 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 spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order , in contrast to the 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 that arises in the representation of the disorder as for standard normal. Under an additional regularity assumption on , we further show that the variance is of order at most . 

Abstract For $$p\geq 1$$ and $$(g_{ij})_{1\leq i,j\leq n}$$ being a matrix of i.i.d. standard Gaussian entries, we study the $$n$$-limit of the $$\ell _p$$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $$p=\infty $$, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $$1\leq p<2$$ and $$2<p<\infty .$$ For the former, we compute the limit of the $$\ell _p$$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $$n^{-1}$$.