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Random matrices and spectral statistics
Connect random entries to global spectra, fluctuations, eigenvalue repulsion, and high-dimensional applications.
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Module 01 research
Random matrices and spectral statistics
Connect random entries to global spectra, fluctuations, eigenvalue repulsion, and high-dimensional applications.
- 01
Wigner ensembles and moment expansions
Track variance normalization and closed-walk contributions before taking a spectral limit.
29 min - 02
Semicircle law and Stieltjes transforms
Choose a consistent resolvent sign and recover a global spectral density from the correct analytic branch.
30 min - 03
Wishart matrices and Marchenko–Pastur law
Normalize rectangular covariance spectra and keep the aspect ratio and zero atom attached to the chosen matrix dimension.
30 min - 04
Resolvents and self-consistent equations
Use exact inverse identities to organize spectral estimates before making probabilistic approximations.
32 min - 05
Concentration and nonasymptotic norm bounds
Turn Gaussian singular-value estimates into finite-dimensional spectral and conditioning guarantees.
30 min - 06
Gaussian ensembles and joint eigenvalue densities
Derive eigenvalue repulsion while keeping variance, symmetry, ordering, and normalization explicit.
33 min - 07
Orthogonal polynomials and determinantal kernels
Build a finite-rank projection kernel and distinguish factorial correlation functions from probability densities.
35 min - 08
Bulk and edge universality
Compare microscopic spectral limits while stating the entry assumptions, symmetry class, and averaging involved.
38 min - 09
Spiked models and BBP transitions
Separate additive and covariance spikes and compute the emergence of a spectral outlier and eigenvector alignment.
36 min - 10
Free probability and asymptotic freeness
Identify the mixed-moment rule behind independent large Gaussian matrices and distinguish it from classical independence.
38 min