SYK Model and q-Brownian Motion

Recently I became aware of the so-called Sachdev-Ye-Kitaev (SYK) model, which has attracted quite some interest in the last couple of years in physics, as a kind of toy model for quantum holography. What attracted my attention was the fact that in some limit there appears a q-deformation of the Gauss distribution – the same one which also showed up in my old papers with Marek Bozejko and Burkhard Kümmerer on non-commutative versions of Brownian motions, see here and here. Whereas in the SYK context there is usually only one limit distribution, in our non-commutative probability context we usually have the multivariate situation with several random variables (corresponding to the increments of the process). Thus I wanted to see whether one can also extend the calculations in the SYK model to a multivariate setting. This is done together with Miguel Pluma in our paper The SYK Model and the q-Brownian Motion. It turns out that one gets indeed q-Gaussian variables corresponding to orthogonal vectors for independent SYK models.

It is not clear to me whether such independent copies of SYK models have any physical relevance. However, there have recently been some papers by Berkooz and collaborators, here and here, where they calculated the 2-point and the 4-point function for the large N double scaled SYK model, by using also essentially the combinatorics of such multivariate extensions.

Those calculations are quite technical and not easy, and it seems to be unclear whether one can get a final analytic result. This seems to be related to our problems of doing any useful analytic calculations with the multivariate q-Gaussian distribution, which is one of the main obstructions for progress on free entropy or Brown measures for the q-Gaussian distributions. (Okay, there has been some progress via free transport by Alice Guionnet and Dima Shlyakhtenko, but this is quite abstract without concrete analytic formulas.) It would be nice (and surely helpful) if we could get some more concrete description of the operator-valued Cauchy transform of the multivariate q-Gaussian distribution.

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