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Program:
- Introduction: applications of random matrices in various domains of mathematics (probability and statistics, statistical physics, number theory)
- Wigner random matrices, concentration inequalities and applications in random matrix theory, the semi-circle law and the Marcenko-Pastur law
- Invariant ensembles of random matrices, determinantal and pfaffian point processes, exact formulas for eigenvalue statistics.
- The Selberg integrals
- Universal limit laws for the spectrum of random matrices: Sine process in the bulk, Airy process at the edge, Tracy-Widom distribution for fluctuations of the maximum
- Potential theory and equilibrium measures. Dyson-Schwinger equations
- Expansion of moments and cumulants in the large size limit.
- If time permits: Introduction to free probability, R-transform and applications, a glimpse on random tilings
Prerequisites
- Ana I-II-III, Stochastik 1 + notions of convergence of random variables from Stokastik II (some reminders will be provided)
- Some knowledge of Funktionentheorie will be useful.
Several topics of independent interest will be covered and applied along the way: concentration inequalities, the Phragmen-Lindelöf principle, trace class operators, potential theory, etc.
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