Chapters:
0:00 Intro
2:21 What is RMT
7:12 Ensemble Averaging/Quantities of Interest
13:30 Gaussian Ensemble
18:03 Eigenvalues Repel
28:08 Recap
29:08 Three Surprising Coincidences
32:44 Billiards/Quantum Systems
36:00 Reimann Zeta

~~~~~~~~~~~~~~~~~~~~~~~~~ Errata + Clarifications ~~~~~~~~~~~~~~~~~~~~~~~~

05:01 The covariance matrix should be denoted by C instead of X.

06:07 To be clear: H_ij gives what physicists call the transition amplitude to go from state j to i. To get the actual transition probability, one takes the magnitude squared, P(i to j) = |H_ij|^2. Because H_ij and H_ji are complex conjugates of each other, both entries encode the same information and yield the same probabilities, so the probability is symmetric P(i to j ) = P(j to i).

06:17 When we say the eigenvalues are interpreted as energies, the list on the RHS really should read (E_1, E_2, … ).

06:18 The equation in the bottom right should be H_ij = H_ji*.

07:00 To be clear, we demand the covariance matrix C to be symmetric such that C = C^T and the Hamiltonian matrix H to be Hermitian such that H = H^\dagger (physicists’ notation) or H = H* (mathematicians’ notation).

16:48 In the expression for Z, the whole argument of the exponential should be multiplied by a factor of N, and the offdiagonal entries should have a factor of 2.

17:20 See previous remark about H_ij encoding transition probabilities.

33:05 For the Sinai billiard: To be clear, the potential is infinity outside the walls of the square arena and in the circular barrier in the center, but is zero on the interior of the arena.

~~~~~~~~~~~~~~~~~~~~~~~ References [in construction] ~~~~~~~~~~~~~~~~~~~~~

Analytic continuation in the Riemann zeta function: 1) video by 3b1b 2) notes from Math 259 at Harvard.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Credits ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Animation team: Artin Kim, Peter Gao, Adin Ackerman, Abrun Nereim.
Original music by: Abrun Nereim, Adin Ackerman.
Image for title sequence: Cafe Venetia.

~~~~~~~~~~~~~~~~~~~~~~~~~ Acknowledgements ~~~~~~~~~~~~~~~~~~~~~~~~~

This video wouldn't have been possible if it wasn't for the animation team!

CJ would also like to thank the following:
D.Z.S. for introducing and enthusiasting me to/about quantum chaos and RMT.
S.S. for beautiful class on RMT.
Livan, Novaes, and Vivo for these beautiful and accessible introductory notes: https://arxiv.org/pdf/1712.07903.pdf.