In the recent post of a similar title I mentioned some papers which related physics problems (eigenstate thermalization hypothesis or Open Quantum SSEP) with free probability. Let me point out that the title of the preprint by Hruza and Bernard has been changed to “Coherent Fluctuations in Noisy Mesoscopic Systems, the Open Quantum SSEP and Free Probability” and that there are some new and follow up preprints in this directions, namely “Spectrum of subblocks of structured random matrices: A free probability approach“, by Bernard and Hruza, and also “Designs via free probability“, by Fava, Kurchan, Pappalardi. In all of them free cumulants and their relations to random matrices play an important role. Not too surprisingly, I find this very interesting in general, but also in particular, as during my voyage in the machine learning world I became a bit obsessed with the fact that free cumulants are given by the leading order of classical cumulants of the entries of unitarily invariant matrix ensembles (with a “cyclic” or “loop” structure of the indices). This seems to be highly relevant – though, at the moment I am not actually sure for what exactly.
Anyhow, if anybody is interested in this, in the last lecture of my machine learning course I give a very high level survey on these relations, and in the video on Gaussian equivalence principle in the same course I talk about a more concrete model of this in the random features model context.
Free Probability Theory 