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: arxiv.org/pdf/1712.07903.pdf.

[@CharlesPanigeo]

Great video! My undergraduate thesis had to do with random matrices. My question was, "given a NxN matrix with each entry is an iid standard normal distribution, what is the probability distribution of the determinant of that matrix?" This is clearly a Gaussian ensemble. I did find a solution, but only in terms of Meijer G functions. I actually found a few papers on the topic. 1 paper from the 60s got the solution in terms of a Mellin-Barnes type integral. Another paper from 2014 i think got the solution in full. My thesis was to synthesis ideas from both papers to create a new proof that was simpler.

[@Ron_Shvartsman]

wonderful! do you have a link to your thesis?

[@user_2793]

I want to do my undergraduate thesis on RMT, could you possibly give me a reading list you used to get familiar with it? I have a fairly strong undergraduate physics background in Stat mech and QM, Linear algebra (at the level of Hoffman and Kunze) and I am not uncomfortable with proofs.

[@web2wl00p]

This was awesome! With a lot of higher math / physics content here on KZbin there is a tendency towards (over)simplification ... this video does not do this and that is very much appreciated. Best of luck for #SoME2

[@Bemajster]

Deadline for SoME2 was 15 of August...

[@lexinwonderland5741]

WOW! This was fantastic!! I really hope you continue with the series, I intend to watch every single video!

[@glennhoetker2721]

Just discovered this awesome channel. Very nice video. You have a real touch for presenting complex material in an engaging way.

[@kafka8103]

what a strong start to new math channel. hope you could make more soon.

[@tanchienhao]

awesome video! ive always wanted to study RMT but this is the first video i know of that introduces it in a beginner friendly manner!

[@thoppay76]

I am so glad to have stumbled to this video. Amazing content expressed in a beginner friendly manner. Thanks a lot.

[@ShinjiCarlos]

I have watched just 7:00 minutes so far and I am already loving every second of the video. Even the choice of soundtrack fits perfectly.

[@Jinbmt656]

Really nice introduction, I can't wait to see what you come up with in the next videos :) I'm a physicist familiar with RMT myself, it's great you're able to make it more accessible on KZbin! If I had one little issue with the format, it's that the sound isn't so great at times, but I guess it'll improve over time. Good luck, I'll be following you closely!

[@radelfalcao9327]

Awesome video. Top quaility content. Started self studying on random matrix theory.

[@robertmines5577]

I've been looking into the Tracy Widom distribution (casually) for a few years now and have struggled to find introductory material on this subject. I'm extremely excited about this upcoming lecture series on random matrices because it's going to help me so much to finally understand this distribution!

[@frozencryo3148]

As someone who got to play around with random matrices in their dissertation work and the joys of them through a physics degree, this video is truly a great watch! One of the best SOME2 vids I’ve seen!

[@MATHsegnale]

Wow!! Very nice video! I was just thinking some days ago I would like to learn a bit about the theory of random matrices. Well done, and thank you!

[@ryderbrooks1783]

It's often the case that the simple/intuitive/lateral aspects are what gets lost in self learning on the internet. Your work here is extremely valuable. Thank you

[@johnchessant3012]

Very interesting topic! The Hilbert-Polya conjecture is really fascinating; it implies RH, which means a potential avenue for proving RH would go through quantum mechanics. Also iirc Montgomery and Dyson met by chance at an interdepartment tea time at IAS; to think, such a cool discovery resulted from such an unlikely encounter. Anyway, great video (and very creative channel name)!

[@ndmath]

Great video. I am sure this channel will grow to something big.

[@TheR971]

"This is hopefully the beginning of a series" instant sub.

[@fibbooo1123]

Awesome video! I'm looking forward to more!

[@lambdapractitioner835]

This is an amazing video. This is one of the only videos I have seen that explains the topic without requiring one to watch several hours of a lecture series to even get a basic understanding. Thank you. Additionally, your channel description is intriguing. Two baristas who make mathematical physics videos .... there is an interesting story for sure.

[@rachelmcmaster7236]

Thankyou for helping me write my Random Matrix Theory section for my Masters thesis on Quantum Chaos!

[@PastaPics]

If I understood matrices a bit better I would be crying tears of joy at how beautiful this is

[@ryderbrooks1783]

Wonderful. I look forward to more videos. Thank you

[@TheJara123]

Super presentation....amazing....please keep posting

[@gammaknife167]

My Master's essay was on the question of the "Longest increasing subsequence of a random permutation of the numbers from 1 to n". It turns out that sequences of numbers from 1 to n can be bijected with objects called Young tableaux in such a way that the length of the longest row of the tableau is the length of the longest increasing subsequence. One can use this bijection to show that the longest increasing subsequence asymptotically has length 2sqrt(n), but what's even more remarkable is that, upon scaling, the variation of the length of the longest increasing subsequence from this 2sqrt(n) quantity follows that of the Tracy-Widom distribution. More universality! The proof is so incredibly gross and trundles through pages and pages of DEs, but it's a brilliant result! Dan Romik has written a great book on the subject, "The Surprising Mathematics of the Longest Increasing Subsequence".

[@apophenic_]

I like your words, magic man. For real though, can you explain to a layman "why" those statements are so remarkable? I understand there's some pattern there and they seem to have some relationship with others, but why is that "remarkable"? What does that mean to you to make it so?

[@gammaknife167]

@@apophenic_ Yeah I'm sorry. I was quite lazy. I haven't defined carefully anything and I did so because I knew it would take a long time and a lot of writing to explain properly. But to answer your question, this is remarkable because it's another instance of universality as described in the video, that's all. Its in a field you wouldn't expect it in - that of combinatorics (the study of counting things that are difficult to count). And why should people care?... I don't know, most mathematical results these days aren't directly useful. Most pure mathematicians are just proving results in the hopes someone else has some use for it in the future. Surely there's SOME use for knowing how long of a subsequence you can find almost always in a random sequence of numbers from 1 to n. If you're interested in some actual definitions you can follow, again, go read Dan Romik's book. Alternatively, go watch the sofa problem on numberphile because that's also Dan Romik talking, and he's just a cool guy :)

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video.

[@symbolspangaea]

Thank you for an amazing 41 minutes!

[@spde]

I hope you will have time to make more videos - I absolutely love your presentation style 😍👏

[@not_David]

I won't be able to watch this one in one go but from what I've seen so far this might just be my favourite SoME2 entry. Please continue making more!

[@krinko547]

I guess you haven't gotten to the half-way pt, it gets a little rough ;'). But thanks that means a lot!!

[@not_David]

@@krinko547 Well I just finished it and I have to say I disgree - It was still great! Excited to see more!

[@PixelSergey]

This is so cool! We only covered random vectors in probability but our professor mentioned the concept of random matrices and more abstract random variables. Really cool to finally see them in action ❤

[@fernandojimenezmotte2024]

I love your video about Random Matrix Theory specially the spectrum distribution of eigenvalues in correlation matrices. I am a scientist, Electrical Engineer and it is great to see how physicists apply RMT to handle for example the complexity of the Hamiltonian of the Uranium nucleus. I will be alert to your next great video. Thank You

[@symbolsforpangaea6951]

This video is awesome!!! Thank you!

[@udaybhosale]

Very interesting one. Need more such videos:)

[@Walczyk]

amazing video, needs more love

[@hoseinzahedifar1562]

Thank you... Amazing presentation ❤❤💗💗

[@metanick1837]

Awesome Video!

[@prwi87]

Can't wait for more!

[@shubhamdawda7288]

Truly enjoyable, hope to see more on this! Maybe demonstration of applicability of RMT to more problems, such as finance?

[@himanshusingla9379]

🎯 Key Takeaways for quick navigation: 00:00 📚 Atomic nuclei, chaotic billiards, and the Riemann Hypothesis are seemingly unrelated, but they're all connected through random matrix theory (RMT). 03:08 🧮 Random Matrix Theory (RMT) is a model for matrices with random entries, used to describe various systems such as atomic nuclei and chaotic billiards. 08:22 🔍 RMT focuses on computing averages of unitarily invariant quantities, often related to the eigenvalues of matrices. 14:03 📜 The Gaussian Ensemble is a fundamental random matrix model, providing simplicity for calculations and capturing physical behavior. 17:48 🧪 Studying eigenvalue spacings reveals that correlated eigenvalues differ from uncorrelated ones, shedding light on randomness and structure in matrices. 21:58 📊 The probability distribution of eigenvalue spacings in correlated events or eigenvalues follows a Gaussian distribution multiplied by a linear term in the spacing. 24:17 📐 The distribution of spacings suggests that for small values of the spacing, it's proportional to the spacing itself, resembling a distance from the origin in polar coordinates. 26:23 🧮 The derived distribution for correlated eigenvalues matches the Wigner surmise and is proportional to `s * e^(-s^2)`, showing that eigenvalues repel each other as they get closer. 27:29 💥 The key difference between Poissonian and Wigner-Dyson distributions lies in the limit as the spacing goes to zero; the latter's limit is zero, representing repulsion, while the former's limit is one. 35:46 🎭 Complex functions like the Riemann Zeta function's zeros display the same spacing distribution as eigenvalues of random matrices, leading to insights about prime number distribution and quantum chaos

[@Number_Cruncher]

Very nice overview. Thanks so much.

[@andriypredmyrskyy7791]

I'M HERE FOR IT

[@cerioscha]

Great video !

[@Heymisterbadguy]

Oh god I'm excited to watch this. My bachelor's scientific initiation (and also the course finish work) was about Random Matrices. Absolutely no one in my university (UNICAMP) was working with them, including my mentors, so I had to study them on my own, which was hard but exciting. I miss research.

[@ep9196]

Great Topic !

[@user-wr4yl7tx3w]

This is really well explained and interesting.

[@davidjohnston4240]

We can use random square matrices with elements in GF(2) for 2 source entropy extractors, although there are more efficient (read that as sparse) number theoretic matrix constructions for this, like in the 2-EXT algorithm. A nice property of that, besides punching through the 50% barrier for single source extractors, it is a quantum secure extractor.

[@kaushalgagan6723]

It's great content. Thanks for nice explanation : )

[@tcaDNAp]

You know it's serious when it switches to a second narrator! And I'm gonna be watching for that distribution; it immediately reminded me of potential energy in chemical bonds! 🤯

[@athreyamurali1439]

Great video, you mentioned links to additional reading in the description, but I only see the timestamps for the video chapters.

[@masterludovicus802]

What a great video. Sadly this is the only one. Can I ask what happened that you stopped making this amazing content?

[@ShinjiCarlos]

About the statement around 6:00 minutes, actually the Hamiltonian needs to be hermitian, since hermitian operators have real valued eigenvalues. This is a requirement of one of the postulates of Quantum Mechanics: "Measurables are represented by operators and their possible outcomes are their respective eigenvalues." Therefore, their eigenvalues need to be real valued. In other words, the operators need to be hermitian. That is: Such an operator needs to be equal to its complex conjugated transpose. A proof connecting these two statements may be found in Quantum Mechanics, a book by Shankar, in a very simple way.

[@crichigno]

Excellent!

[@maguedoudasamiabelhaddad2183]

Brilliant !

[@onebronx]

Very interesting! 27:30 - the Wigner-Dyson distribution looks very similar to the 2D Mazwell-Boltzman distribution, just without the (m/kT) scale factor.

[@jlaxgang7233]

mathematics is my God and my staff I pray like this and break minds in half

[@NinjaOfLU]

I hope this doesn't get screwed over by the fact that it has some prior knowledge assumptions. I kept expecting you to take a step back and explain what an eigenvalue was, or something, but I appreciate you choosing your audience carefully enough to know that wasn't necessary, and sticking to it!

[@amoswittenbergsmusings]

Wow, I love this and I subscribed. Suggestion: drop the music. It does not add anything and tends to distract.

[@stevenytcx]

Keep going!

[@TrailersReheard]

I think you're going at the perfect pace

[@fabiangn8022]

Gracias ❤️👏🏽

[@vilhelmlarsen9565]

Amazing

[@fkeyvan]

Great work. Thank you. How does RMT relate to Brownian Motion?

[@valor36az]

Please continue to make more videos

[@hareecionelson5875]

all hail the youtube algorithm I just got done coding a matrix multipler and a determinant calculator in python, which also necessitated coding a random matrix generator to generate some samples matrices to feed into the calculators. I even used a -10 - 10 range, but with a small bias towards 0s (test the efficiency of the determinant calculator, it should be able to start from the row or column with the most 0s to avoid extra work.) and a switch to allow complex numbers to occasionally pop in. My obsession with coding matrices has been vindicated. I need to tell my friends.... oh wait they've gone.

[@nativesun7661]

Great content - I’d definitely recommend going slower, breaking things up into separate videos and emphasizing/repeating key points. You have such a wealth of useful information- give it to the people in digestible chunks. But clearly very thoughtful & informed presentation.

[@midou6104]

Why you stopped after just one video 🤨🤨🤨 You really great, plz continue we don't have a lot of content related to random matrix or random NLA

[@turdferguson3400]

Your channel name is very clever

[@eqwerewrqwerqre]

This is a huge amount of information thrown at me very very fast. It felt like you guys were trying to speak as fast as possible and the animations kept changing before i could understand what they were trying to convey. It's an extremely accurate and complete (i assume) introduction to random matrix theory but most of the people i see in the comments who get it are people who already work with rmt. For someone who has all the linear algebra background but has never seen this subject before i'd need to watch this at half speed at most and rewatch the animations to try to suss out what was being communicated. I think you guys did a great job for an early go at this but damnnn. Just go a little slower. You don't need to fit the entire thing into a single video. This video could be twice as long and still difficult. It's really cool stuff, you're doing great

[@davidmurphy563]

Ok, I'm going to have to sit down with a pen, paper and work through this. Actually, what am I saying? I'll put it in a games engine like a normal person. I'll need to brush up on the determinants (that's the area you use to work out the cross product if I remember right) eigenvectors (that's the direct transformation axis, I always forget that one) and a dozen other things. They were almost all terms I had heard of which is encouraging and what was being crunched was pretty clear. Love that this is "the basics" for a physicist but I'm seeing it as a sheer face to climb with a few potential handholds I might be able to hang on to if I sweat blood.

[@wanfuse]

very good work, said very well. I would suggest that with NN's one might be able to determine a halucination from a real thought or to determin if NNs are human aligned in the NN by looking at the distributution and see how closely it matches the original data. it wont be exact but it would be countable , I have done something similar with my bag of marbles sort , on sets that are in the trillions you can select 10 percent of the binary values randomly and get back the original, with a quadrillion even better ratios. you need a perfect random generator though at some point to insure that random selection is truely made. I do not know if cryptographic random is enough though. Any thoughts?

[@spaceisawesome1]

Very cool video! Are you aware on applications for RMT as it applies to non negative matrix factorization problems? I am interested in working with matrices that come in this form and are most likely non square

[@juvenwang]

Thank you for this --- what is the source of the background music?

[@aeebeecee3737]

I subbed your channel

[@ryanlangman4266]

This was a great video. The topic was super interesting, and I learned a lot. I think there’s a small typo at 37:15 in the product form of the Reimann Zeta function. The k in the exponent of p should be multiplied by z so that it’s the product over primes of 1/(1-p^(-z)). As its stated there seems to be no z dependence on the right hand side of the equation. I haven’t worked with this function much though, so I could be wrong. I’m also curious about the Hamiltonian. The Hamiltonian operator is usually infinite dimensional. So, does that change the calculation in any way? How do we deal with infinite dimensional random matrices? Especially when we consider the uranium atom. How could we possibly go about evaluating anything about it, since it’s both infinite dimensional and has massively complicated interactions? Also how would we use computers to work with infinite dimensional matrices? Do we somehow specify a basis that will approximate the answer with only a finite number of dimensions, or how is that done?

[@NikolajKuntner]

thanks

[@usptact]

Great video! I even managed to understand good portion of it! Tongue in cheek: what about Random Tensor Theory? ;) Are there any such systems where random tensors would arise?

[@cryoshakespeare4465]

Aren't matrices just a subset of tensors? (ie. 2d versus Nd). Seems like the theory would be similar.

[@whoknowsnubby]

This is sick

[@134ayush]

When will next set of videos be released?

[@telotawa]

the equations for perfectly elastic collision can be thought of as a symmetric matrix

[@tolkienfan1972]

Very cool. What happened to the audio at 30:20??

[@smolboi9659]

22:20 That's the rayleigh distribution. That is the distribution of the magnitude of a complex number with i.i.d guassian real and imaginary parts with 0 mean.

[@AkamiChannel]

Nice use of manim

[@jimgorlett4269]

i need to brush up on a lot of math before i rewatch this

[@Unaimend]

Will you continue this series?

[@HUEHUEUHEPony]

Impressive, now explain it to me as if I was four.

[@anonym0usplatypus]

Why do you call it the wigner-dyson distribution? I've always seen this denoted as the Rayleigh distribution

[@wolliwolfsen291]

I’am waiting for a follow up video

[@samosavaglio2141]

There is a z missing when you write the euler product for the zeta

[@joyboricua3721]

Through 3/5 of it & my impression summarizes in this opinion: What a tasteful way to b-slap my left lobe!

[@ad2181]

Thank you your RMT presentation. For the Elderly please increase the audio recording level. Use color pairs that are high contrast and easy on the eyes.

[@krinko547]

Will bear that in mind for the coming video, thanks for your feedback!

[@deniskochan8230]

Can someone explain what is s, P(s) and q(s) at 19:12? q(s) was defined via P(s), but P(s) and s are not clear to me. Thanks in advance

[@haidaralhassan4621]

Hello it’s my first time in this channel 🙋🏻 I didn’t complete the video yet I’m just at the start but I have a small request 😅😅😅😅 Why the solid black screens? 😭😭😭 they stay for a very long time and make me super anxious thinking the display is off 😭😭😭 I suggest putting something that moves as a place holder especially when the black screen is gonna take so long like from 1:18 to 1:38 that’s almost 20 seconds of solid black screen 😅😅 I hope you find this constructive as I’m sure the rest of this video is gonna be so enjoyable to me 😊

[@krinko547]

Thanks for the feedback Haidar! We were scrambling to get it out in time for a deadline, so it does get very Frankenstein-esque at times, and especially mid-way. We're trying to fix these things and hopefully upload a cleaner version!

[@XORfun]

Thank you for the great work! Don’t forget to subscribe

[@Achrononmaster]

On the Riemann zeta function, a tip I'd give any young math whizz is to ditch matrix algebra and use the proper real geometric algebra. Every matrix is a multivector in a geometric algebra, and that's the "proper" setting for studying the Riemann Hypothesis. Most number theorists do not know this, so you can get a head start. Studying the zeros the amounts to studying when a infinite sum of rotors passes through zero. It's mostly a geometry puzzle, not just a number theoretic puzzle.

[@notnilc2107]

i've seen the word "hermitian" a lot and i never understood what it meant or why it was important.

[@vinniepeterss]

❤❤

[@fullfungo]

2:00 I am confused. Why did you put Markov chains in the Biology corner?

[@ayushsharma8804]

Computational genomics and proteomics uses Markov chains very heavily, probably it's biggest application

[@user255]

37:36 The lower equation, where is the z in right hand side?

[@krinko547]

You are right -- the z is missing. It should be multiplying the power of k in the sum on the RHS, i.e. \zeta(z) = \sum_{n=1}^{\infty} 1/n^z = \prod_p 1/(1-p^{-z}) = \prod_p [ \sum_{k=0}^{\infty} p^{-zk}], where p is the product over all primes, and in the last step we've just used the binomial expansion 1/(1-x) = \sum_{k=0}^{\infty} x^k. Thanks for catching this!

[@josephdays07]

I have developed a new theory, I have called Partitions Trigonometric and I have discovered something amazing. I can do X Rays with these equations applied to Z Riemann Equation.

[@josephdays07]

kzbin.info/www/bejne/eHPdi6djg7yXp6M

[@nestorv7627]

RMT is the powerhoucell of the cell

[@jackalvarez7428]

This would be so much better if it weren’t narrated by TTS 😅