Random Matrix Theory: A tale of drums, earthquakes and mobile phones
Dr Mario Kieburg from the School of Mathematics & Statistics at the University of Melbourne presented the topic of Random Matrix Theory on Tuesday 29th September. You can view the recording and see the answers to the additional questions below.
Title – Random Matrix Theory: A Tale of Drums, Earthquakes & Mobile Phones
Date – Tuesday 29th September
Time – 4 - 5pm
Unanswered questions from the audience
- For dimension 1, I can understand what is k that is amplitude but in a rectangular case how we count k_x and k_y?
Answer - k_x counts the number of peaks and troughs in the x-direction and k_y does it in the y-direction. Note, that this does not work anymore for irregularly shaped drums. Then it is only an ordering of the speed of the vibration of the single resonance modes from slow to fast.
- Where would the random matrix entries come from (for example, in the case of the Mexico City drum modelling)? Are the results of this sort of analysis used in city planning?
Answer - The random matrix entries do not come from a physical system. They are replacements for the physical world. This is the reason why they cannot explain or predict an exact pattern of destruction. Hence, you cannot explicitly use a random matrix for city planning. Random matrices describe the statistical properties of this pattern. Say once you found an area of large destruction, you can answer how likely it is to find at a certain distance another one. Nevertheless, the explicit numerical computations of the waveforms are indeed exploited for future planning.
- Is there a paper on the similarity of the statistical properties?
Answer - A good overview of random matrix theory and especially of their applications gives the book "The Oxford Handbook of Random Matrix Theory" by Akemann et al. A bit older but easily accessible work is the one by Guhr et al. which you can find under the open link arXiv:cond-mat/9707301. This review is also on the topic of the public talk.
- What do the values in the random matrix represent? With the drum example, would it be the height of the membrane at different coordinates?
Answer - Yes, the entries can be interpreted as heights of the membrane at specific positions.
- Does the strength of the earthquake affect the location of the peaks much?
Answer - Yes, it is very important in what way and with what strength the earthquake hits the "sandy lake" underneath Mexico city. The pattern of destruction certainly changes. However, the statistics of the peaks and troughs do not. The latter is exactly what can be assessed and described with the random matrix approach.