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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 $$n^{-1}$$.
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Report on BASICS: Lesson Plan on Aerosols and Infection
- Award ID(s):
- 1660924
- PAR ID:
- 10249941
- Date Published:
- Journal Name:
- The Biophysicist
- ISSN:
- 2578-6970
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.35459/tbp.2021.000176
