Ioana Dumitriu (4.2) Wigner random matrices and the Semicircle Law, part 4.1
Channel: IAS | Park City Mathematics Institute | PCMI
Date: 2017-06-30

Links to lecture notes found at https://pcmi.ias.edu/sites/pcmi.ias.edu/files/Ioana%20Dumitriu%20Lecture%204.pdf

Ioana Dumitriu, University of Washington
Eigenvalues, graphs, and Catalan numbers: the semicircle law and beyond.

The most basic, and hence arguably most famous result in random matrix theory is the (Wigner) semicircle law. Simply stated, for any Wigner symmetric/Hermitian random matrix with independent, identically distributed entries of mean 0 and variance 1, the distribution of a randomly-chosen eigenvalue converges to the (normalized) semicircle. This result is true regardless of the entry distribution (with only very minor conditions). The proof involves linear algebra and combinatorics: in particular, looking at traces of powers of the matrix and counting walks on the complete graph to obtain Catalan numbers. We will prove this result and then delve into why much more can actually be shown. From the fact that the shape of the histogram for all eigenvalues is semicircular, to investigating the fluctuations from the semicircle, we will explore the surprisingly stable structures that emerge from the spectrum of large random matrices.

Presented at the 27th Annual PCMI Summer Session, Random Matrices, held June 25 – July 15, 2017. The residential, three-week Summer Session is the flagship activity of the IAS/Park City Mathematics Institute (PCMI).

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