Spherical Integrals of Sublinear Rank

Abstract

We consider the asymptotics of $k$-dimensional spherical integrals when $k=o(N)$. We prove that the $o(N)$-dimensional spherical integrals are approximately the products of $1$-dimensional spherical integrals. Our formulas extend the results for $k$-dimensional spherical integrals proved by Guionnet and Maïda in [29] and Husson and Guionnet in [34] 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.

Corrected the argument in Section 3.1. Removed an assumption in Proposition 2.8. Corrected typos

References in corpus (5)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• On the overlap in the multiple spherical SK models
• Character expansion method for the first order asymptotics of a matrix integral
• The Crisanti-Sommers Formula for Spherical Spin Glasses with Vector Spins
• Optimal denoising of rotationally invariant rectangular matrices