https://www.chenk.top/en/tags/marchenko-pastur-distribution/Recent content in Marchenko-Pastur Distribution on Chen Kai BlogHugoenWed, 02 Apr 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/14-random-matrix-theory/Wed, 02 Apr 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/14-random-matrix-theory/<p>A million i.i.d. coin flips, arranged into a thousand-by-thousand symmetric matrix, somehow produce eigenvalues that fill a perfect semicircle. A noisy sample covariance matrix that should be the identity instead spreads its eigenvalues across an interval whose width you can predict before seeing a single number. The largest eigenvalue of a Wigner matrix has a tail distribution that turns up everywhere — in growing crystals, in the longest increasing subsequence of a random permutation, in the energy levels of heavy nuclei. <strong>Random matrix theory</strong> (RMT) is the study of why these regularities appear, and how to use them.</p>