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<p>We consider the asymptotics of rank <i>k</i> spherical integrals when <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="440_2025_1402_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(k = o(N)\)</EquationSource> </InlineEquation>. We prove that the sublinear rank spherical integrals are approximately the products of rank 1 spherical integrals. Our formulas extend the results for rank <i>k</i> spherical integrals proved by Guionnet and Maïda in Guionnet and Maïda (J. Funct. Anal. 222:435–490, 2005) and Husson and Guionnet in Guionnet and Husson (ALEA. 19:769–797,&#xa0;2022) which are only valid for <i>k</i> finite and independent of <i>N</i>. These approximations will be used to prove a large deviation principle for the joint 2<i>k</i>(<i>N</i>) extreme eigenvalues for sharp sub-Gaussian Wigner matrices and for additive deformations of GOE/GUE matrices. Furthermore, our results will be used to compute the free energies of spherical SK vector spin glasses and the mutual information for matrix estimation problems when the dimensions of the spins or signals have sublinear growth.</p>

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Spherical integrals of sublinear rank

  • 期刊论文

  • 发表日期：

    2025年8月5日

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  • Jonathan Husson,
  • Justin Ko

摘要

We consider the asymptotics of rank k spherical integrals when \(k = o(N)\) . We prove that the sublinear rank spherical integrals are approximately the products of rank 1 spherical integrals. Our formulas extend the results for rank k spherical integrals proved by Guionnet and Maïda in Guionnet and Maïda (J. Funct. Anal. 222:435–490, 2005) and Husson and Guionnet in Guionnet and Husson (ALEA. 19:769–797, 2022) which are only valid for k finite and independent of N. These approximations will be used to prove a large deviation principle for the joint 2k(N) extreme eigenvalues for sharp sub-Gaussian Wigner matrices and for additive deformations of GOE/GUE matrices. Furthermore, our results will be used to compute the free energies of spherical SK vector spin glasses and the mutual information for matrix estimation problems when the dimensions of the spins or signals have sublinear growth.