2016/03/31 by Joël Bun, Jean‐Philippe Bouchaud, Jean-Philippe Bouchaud +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Applied mathematics #Eigenvalues and eigenvectors #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Multiplicative function #Physics #Pure mathematics #Quantum many-body systems #Quantum mechanics #Random Matrices and Applications #Random matrix #Spectrum (functional analysis) #Statistical physics #Theoretical and Computational Physics #cond-mat.stat-mech #physics.data-an #q-fin.MF
paper · pdf · doi:10.1103/physreve.98.052145
published as Phys. Rev. E 98, 052145 (2018) · 12 pages, 5 figures
arxiv created 2017/07/23 · openalex publication_date 2018/11/29 · arxiv updated 2018/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum thermalization to high-dimensional statistics. We find that the overlaps only depend on measurable quantities, and do not require the knowledge of the underlying ``true'' (noiseless) matrices. We apply our results to the case of empirical correlation matrices, that allow us to estimate reliably the width of the spectrum of the true correlation matrix, even when the latter is very close to the identity. We illustrate our results on the example of stock returns correlations, which clearly reveal a nontrivial structure for the bulk eigenvalues. We also apply our results to the problem of matrix denoising in high dimensions.