MA 360: Random Matrix Theory
Credits: 3:0
-
Wigner’s semicircle law: (a) combinatorial method, (b) Stieltjes’ transform method, (c) Chatterjee’s invariance principle method.
-
Gaussian unitary and orthogonal ensembles: (a) Exact density of eigenvalues. (b) Orthogonal polynomials and determinantal formulas leading to another proof of Wigner’s semicircle law.
-
Tridiagonal reduction for GUE and GOE: (a) Another derivation of eigenvalue density. (b) Another proof of Wigner’s semicircle law. (c) Matrix models for Beta ensembles. (d) Selberg’s integral.
-
Other models of random matrices - Wishart and Jacobi ensembles.
-
Free probability: (a) Noncommutative probability space and free independence. (b) Combinatorial approach to freeness. (c) Limiting spectra of sums of random matrices.
-
Non-hemitian random matrices: (a) Ginibre ensemble. (b) Circular law for matrices with i.i.d entries.
-
Fluctuation behaviour of eigenvalues (if time permits).