## Statistical analysis and universal limit theorems in the theory of random matrices

### Prof. Kenneth McLaughlin

Number of cycles: 1

Second semester

Content:

- Introduction to random matrices.
- First example: Gaussian Random matrices, symmetric case.
- Numerical experimentation.
- Definitions of the basic quantities of study.
- Homework: exact evaluation in the case N=2, proving various pieces of the puzzle.
- Second example: Gaussian Random Matrices, self-adjoint case.
- Numerical experimentation.
- Homework: numerical simulations.
- Third example: Wishart random matrices of various kinds
- Numerical experimentation.
- Asymptotic analysis for N large.
- Exact formulae for N finite.
- Homework: evaluation of some exact statistical quantities.
- Summary of asymptotic results.
- Summary of methods employed for these results.
- The sound and the fury of asymptotic analysis: the case of Hermite polynomials.
- Homework: asymptotic analysis canon
- A few statistical applications (presentations by students).
- Limit Theorems!.
- Recent results and open directions for research.