MATH 689601 Random Matrix Theory II, spring 2018
Course information
 Instructors:
 Dr Michael Anshelevich, Blocker 533d,
manshel AT math.tamu.edu, Office hours Th 12, or by appointment.
 Dr Gregory Berkolaiko, Blocker 625c,
berko AT math.tamu.edu, Office hours M 34, by appointment.
 Lectures: TR 2:203:35, BLOC 605ax
 Course description (first day handout)
 The course does not have a textbook. We will most closely
use

(freely available online) "Free Probability and Random
Matrices", by James Mingo and Roland Speicher, Springer, ISBN
9781493969418.

"Random Matrix Theory: Invariant Ensembles and Universality",
by Percy Deift and Dimitri Gioyev, AMS, ISBN 9780821847374.

"Random Matrices", by Madan Lal Mehta, Academic Press.

(freely available online) "Topics in Random Matrix Theory", by Terrence Tao, AMS.
 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.
Matlab demonstrations
Homework Assignments
 Due Apr 30, target submission 5*.
 (*) Show that for a permutation \(\alpha \in S_n\),
\[\#\{i \in [N]^n : i = i_\alpha\} = N^{\alpha} = N^{\alpha^{1}}.\]
 (*) 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
\[i_{\sigma \rho} = i'_\sigma \quad \Leftrightarrow \quad i_{\sigma \rho \sigma^{1}} = i'.\]
 (*) 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.
 (*) 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.
 (*) Use the Grand Orthogonality Theorem to prove that for any two irreducible representations \(\rho, \rho'\),
\[\frac{1}{G} \sum_{g \in G} \chi_{\rho}(g^{1}) \rho'(g) = \delta_{\rho = \rho'} \frac{1}{d_\rho} I_{d_\rho}.\]
 (**) Prove the relation
\[(\chi_{\rho} \ast \chi_{\rho'})(\pi) = \frac{S_t}{d_\rho} \delta_{\rho = \rho'} \chi_\rho(\pi).\]
 (**) Use the relation in the preceding exercise to prove the SamuelBeukers Theorem.
 Due Mar 22, target submission 7*.
 (*) 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
 \(h_0(x) = 1, \quad h_1(x)=x, \quad h_2(x) = x^21, \quad
h_{n+1}(x) = xh_n(x)  h'_n(x),\)
 \(h_n(x)\) is a monic polynomial of degree \(n\).
 \(h_n(x)\) is even (corresp. odd) when \(n\) is even
(corresp. odd).
 (**) 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!).
 (**) Exercise 5.12 from the notes.
 (*) Exercise 5.16 from the notes.
 Exercise 6.12 from the notes (optional).
 (*) Exercise 6.15 from the notes.
 (*) Exercise 6.17 from the notes.
 (*) Exercise 6.18 from the notes.
 (*) Exercise 6.19 from the notes.
 Exercise 6.24 from the notes (optional).
 (**) Exercise 6.26 from the notes.
 (*) Exercise 6.36 from the notes.
 (**) Exercise 6.38 from the notes. An outline is sufficient.
 (*) Exercise 6.39 from the notes. An outline is sufficient.
 (*) Exercise 6.43 from the notes.
This file was last modified on Thursday, 12Apr2018 13:25:16 CDT.