8 Random Matrix Theory
In addition to non-asymptotic results, we will need asymptotic analysis, which is more delicate. The section is organized as follows:
- 1.
- Density of eigenvalues in classical ensembles of random matrices
- 2.
- Semi-Circle Law and Marchenko–Pastur Law
- 3.
- BBP Transition
- 4.
- CLT for Eigenvalues
- 5.
- Spectrum Separation
- 6.
- Replica Method
This section mainly follows STAT C206B at UC Berkeley, taught by Vadim Gorin [9], and STAT 260 at UC Berkeley, taught by Song Mei in 2021 [14]. I also referred to the book [1].
8.1 Density of Eigenvalues in Classical Ensembles of Random Matrices
8.2 Semi-Circle Law and Marchenko–Pastur Law
8.2.1 Trace Calculation
8.2.2 Marchenko–Pastur Law
8.3 BBP Transition
8.4 CLT for Eigenvalues
8.5 Spectrum Separation
8.6 Replica Method
8.2 Semi-Circle Law and Marchenko–Pastur Law
8.2.1 Trace Calculation
8.2.2 Marchenko–Pastur Law
8.3 BBP Transition
8.4 CLT for Eigenvalues
8.5 Spectrum Separation
8.6 Replica Method