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).

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E-mail: chairman.math[at]iisc[dot]ac[dot]in