In this intro video, we would like to explain first of all what random matrix theory is (linear algebra + probability theory, simple examples, the idea of ensembles). Then we draw the connection to physics, most notably the field of quantum chaos. The story begins in the 1960s with Wigner using random matrices to model the interactions of complex nuclei. His prediction was beautifully matched with spectral distributions obtained in experiments. Fast forward to 2019, when google released their hotly debated „quantum supremacy“ paper. Their alleged demonstration of quantum supremacy relied on the dynamics generated by random matrices. Specifically, it was necessary that the quantum circuits they built in the lab elucidate a random matrix ensemble. There have been a lot of applications of RMT in the years in between, most notably in the study of black holes, in the field of high energy physics, but also for predicting the dynamics of spin chains in condensed matter theory. But RMT <-> quantum chaos is only one link in what is actually a trinity. It turns out that not only atomic nuclei, but also the zeroes of the Riemann zeta function follow a distribution described by the same RMT. The trinity is linked through three very deep mathematical insights: Periodic orbit theory, Gutzwiller’s trace formula, and Montgomery’s conjecture.
We want to expose this intriguing trinity in this video, but also in a more down to earth approach explain how widely RMT is used nowadays in physics. It’s a beautiful, deep and confusing subject that doesn’t have enough introductory material. >
About the authors
We are Tamra and Cheryne, two PhD students in Physics at Stanford University. Our overlapping areas of research are quantum many-body dynamics, quantum chaos, and machine learning. We’ve been wanting to start our youtube channel with a miniseries on random matrix theory for a while actually. But we have amongst us a perfectionist and camera shy person, on top of that we’re usually busy with research and other lovely things, so we haven’t made it to the final steps of video production and uploading yet.
Quick Summary
Target audience for the intro video: anyone.
What we’re hoping they’ll get from the lesson: Know what a random matrix theory is and appreciate how multi-faceted and relevant for present-day science they are.
Target audience for the miniseries (this is not for the SoME submission anymore): Undergraduate and graduate students with some knowledge of maths. The more advanced mathematical or physics concepts, like saddle points or Feynman diagrams, we would like to explain in short complementary videos. Note that the diagrammatic expansion in RMT is actually due to t’Hoft, and differs from Feynman’s regular expansion in that the propagators now contain two lines (this is because matrices have two indices). As such, they series is summed using insights from topology in 2d, the important diagrams being the planar ones.
What we’re hoping they’ll get from the lesson: Motivate how to choose a particular random matrix theory to match with a physical model. Here symmetries play a crucial role.Knowing how to get ones hands dirty and compute physically relevant quantities in a chosen RMT, most importantly the density of states and the level spacings. Explain the different mathematical paths one can take to arrive at the same results (saddle point approximation, Feynman diagrams, orthogonal polynomials), some of them being large N (N=matrix dimension) approximations, others being exact. This should give a deeper appreciation for the unsolved mystery of why random matrices can model physical Hamiltonians so successfully, for while they produce similar results, the Hamiltonians describing physical systems differ crucially in that they contain correlations between their eigenvectors.
Target medium
Yes we are aiming for a video! Hoping for someone who is eloquent with simulations, e.g. can make simple cartoons, plots, transitions. It’s not geometry so not too detail-oriented.
More details
We do have an outline on overleaf, willing to share after we get in touch via email :)
In this intro video, we would like to explain first of all what random matrix theory is (linear algebra + probability theory, simple examples, the idea of ensembles). Then we draw the connection to physics, most notably the field of quantum chaos. The story begins in the 1960s with Wigner using random matrices to model the interactions of complex nuclei. His prediction was beautifully matched with spectral distributions obtained in experiments. Fast forward to 2019, when google released their hotly debated „quantum supremacy“ paper. Their alleged demonstration of quantum supremacy relied on the dynamics generated by random matrices. Specifically, it was necessary that the quantum circuits they built in the lab elucidate a random matrix ensemble. There have been a lot of applications of RMT in the years in between, most notably in the study of black holes, in the field of high energy physics, but also for predicting the dynamics of spin chains in condensed matter theory. But RMT <-> quantum chaos is only one link in what is actually a trinity. It turns out that not only atomic nuclei, but also the zeroes of the Riemann zeta function follow a distribution described by the same RMT. The trinity is linked through three very deep mathematical insights: Periodic orbit theory, Gutzwiller’s trace formula, and Montgomery’s conjecture.
We want to expose this intriguing trinity in this video, but also in a more down to earth approach explain how widely RMT is used nowadays in physics. It’s a beautiful, deep and confusing subject that doesn’t have enough introductory material. >
About the authors
We are Tamra and Cheryne, two PhD students in Physics at Stanford University. Our overlapping areas of research are quantum many-body dynamics, quantum chaos, and machine learning. We’ve been wanting to start our youtube channel with a miniseries on random matrix theory for a while actually. But we have amongst us a perfectionist and camera shy person, on top of that we’re usually busy with research and other lovely things, so we haven’t made it to the final steps of video production and uploading yet.
Quick Summary
Target audience for the intro video: anyone.
What we’re hoping they’ll get from the lesson: Know what a random matrix theory is and appreciate how multi-faceted and relevant for present-day science they are.
Target audience for the miniseries (this is not for the SoME submission anymore): Undergraduate and graduate students with some knowledge of maths. The more advanced mathematical or physics concepts, like saddle points or Feynman diagrams, we would like to explain in short complementary videos. Note that the diagrammatic expansion in RMT is actually due to t’Hoft, and differs from Feynman’s regular expansion in that the propagators now contain two lines (this is because matrices have two indices). As such, they series is summed using insights from topology in 2d, the important diagrams being the planar ones.
What we’re hoping they’ll get from the lesson: Motivate how to choose a particular random matrix theory to match with a physical model. Here symmetries play a crucial role.Knowing how to get ones hands dirty and compute physically relevant quantities in a chosen RMT, most importantly the density of states and the level spacings. Explain the different mathematical paths one can take to arrive at the same results (saddle point approximation, Feynman diagrams, orthogonal polynomials), some of them being large N (N=matrix dimension) approximations, others being exact. This should give a deeper appreciation for the unsolved mystery of why random matrices can model physical Hamiltonians so successfully, for while they produce similar results, the Hamiltonians describing physical systems differ crucially in that they contain correlations between their eigenvectors.
Target medium
Yes we are aiming for a video! Hoping for someone who is eloquent with simulations, e.g. can make simple cartoons, plots, transitions. It’s not geometry so not too detail-oriented.
More details
We do have an outline on overleaf, willing to share after we get in touch via email :)
Contact details
cheryne@stanford.edu