## Abstract

For p ≥ 1 and (g_{ij})_{1≤i,j≤n} being a matrix of i.i.d. standard Gaussian entries, we study the n-limit of the ℓ_{p}-Gaussian-Grothendieck problem defined as n max (Equation presented). The case p = 2 corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when p = ∞, 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 ≤ p < 2 and 2 < p < ∞. For the former, we compute the limit of the ℓ_{p}-Gaussian-Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates.

Original language | English (US) |
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Pages (from-to) | 2344-2428 |

Number of pages | 85 |

Journal | International Mathematics Research Notices |

Volume | 2023 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1 2023 |

### Bibliographical note

Funding Information:This work was supported by the National Science Foundation [DMS-17-52184].

Funding Information:

This work was supported by the National Science Foundation [DMS-17-52184]. WKC and AS would like to thank Souvik Dhara for explaining the results of [22] and for a helpful discussion.

Publisher Copyright:

© The Author(s) 2021. Published by Oxford University Press. All rights reserved.

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