## MATH 689-601 Random Matrix Theory II, spring 2018

### Course information

• Instructors:
• Dr Michael Anshelevich, Blocker 533d, manshel AT math.tamu.edu, Office hours Th 1-2, or by appointment.
• Dr Gregory Berkolaiko, Blocker 625c, berko AT math.tamu.edu, Office hours M 3-4, by appointment.
• Lectures: TR 2:20-3:35, BLOC 605ax
• Course description (first day handout)
• The course does not have a textbook. We will most closely use
1. (freely available online) "Free Probability and Random Matrices", by James Mingo and Roland Speicher, Springer, ISBN 978-1-4939-6941-8.
2. "Random Matrix Theory: Invariant Ensembles and Universality", by Percy Deift and Dimitri Gioyev, AMS, ISBN 978-0-8218-4737-4.
4. (freely available online) "Topics in Random Matrix Theory", by Terrence Tao, AMS.
5. Free probability notes from a course taught by M.A. in 2012.
• See also Todd Kemp's notes and Terry Tao's book (with an earlier version linked from his page). Additional resources include the book by Anderson, Guionnet, and Zeitouni (also on the web), and notes by Benedek Valko and Manjunath Krishnapur.

### Course Notes

The notes will be updated cumulatively.

### Homework Assignments

1. Due Apr 30, target submission 5*.
1. (*) Show that for a permutation $$\alpha \in S_n$$, $\#\{i \in [N]^n : i = i_\alpha\} = N^{|\alpha|} = N^{|\alpha^{-1}|}.$
2. (*) Show that for two permutations $$\sigma, \rho \in S_n$$ and $$i, i': [n] \rightarrow [N]$$, $i_\rho = i'_\sigma \quad \Leftrightarrow \quad i_{\rho \sigma^{-1}} = i'$ and $i_{\sigma \rho} = i'_\sigma \quad \Leftrightarrow \quad i_{\sigma \rho \sigma^{-1}} = i'.$
3. (*) Let $$G$$ be a finite group and $$f,g : G \rightarrow \mathbb{C}$$. Show that $\widehat{f \ast g} = \hat{f} \cdot \hat{g},$ where the Fourier transform $$\hat{f}$$ and convolution $$\ast$$ were defined in class.
4. (*) Take $$\mathbb{Z}_3 = \{0, 1, 2\}$$. This finite group has irreducible representations $R_q: \mathbb{Z}_3 \rightarrow \mathbb{C}, \quad R_q(p) = e^{\frac{2 \pi i}{3} p q}.$ Compute the direct and inverse Fourier transforms for this group.
5. (*) Use the Grand Orthogonality Theorem to prove that for any two irreducible representations $$\rho, \rho'$$, $\frac{1}{|G|} \sum_{g \in G} \chi_{\rho}(g^{-1}) \rho'(g) = \delta_{\rho = \rho'} \frac{1}{d_\rho} I_{d_\rho}.$
6. (**) Prove the relation $(\chi_{\rho} \ast \chi_{\rho'})(\pi) = \frac{|S_t|}{d_\rho} \delta_{\rho = \rho'} \chi_\rho(\pi).$
7. (**) Use the relation in the preceding exercise to prove the Samuel-Beukers Theorem.
1. Due Mar 22, target submission 7*.
1. (*) We have defined Hermite polynomials as $h_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} \left[e^{-x^2/2}\right].$ From this definition, prove properties (1)-(3), namely
• $$h_0(x) = 1, \quad h_1(x)=x, \quad h_2(x) = x^2-1, \quad h_{n+1}(x) = xh_n(x) - h'_n(x),$$
• $$h_n(x)$$ is a monic polynomial of degree $$n$$.
• $$h_n(x)$$ is even (corresp. odd) when $$n$$ is even (corresp. odd).
2. (**) Calculate numerically and plot the histogram of zeros of Hermite polynomial of a large order $$N$$. Submit your code printout and a picture. Hint: it is nicer to have $$N$$ fairly large, about 500. But the straightforward way (defining the polynomial, finding its roots) will not work because the coefficients grow too fast. Recast the problem as looking for eigenvalues of a matrix instead (check the typed lecture notes!).
3. (**) Exercise 5.12 from the notes.
4. (*) Exercise 5.16 from the notes.
5. Exercise 6.12 from the notes (optional).
6. (*) Exercise 6.15 from the notes.
7. (*) Exercise 6.17 from the notes.
8. (*) Exercise 6.18 from the notes.
9. (*) Exercise 6.19 from the notes.
10. Exercise 6.24 from the notes (optional).
11. (**) Exercise 6.26 from the notes.
12. (*) Exercise 6.36 from the notes.
13. (**) Exercise 6.38 from the notes. An outline is sufficient.
14. (*) Exercise 6.39 from the notes. An outline is sufficient.
15. (*) Exercise 6.43 from the notes.