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Random Matrices in Unexpected Places: Atomic Nuclei, Chaotic Billiards, Riemann Zeta

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 off-diagonal 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.