Amaury Freslon has put a pre-print on the arXiv, Cut-off phenomenon for random walks on free orthogonal quantum groups, that answers so many of these questions, some of which appeared as natural further problems in my PhD thesis.
It really is a fantastic paper and I am delighted to see my PhD work cited: it appears that while I may have taken some of the low hanging fruit, Amaury has really extended these ideas and has developed some fantastic examples: all beyond my current tools.
This pre-print gives me great impetus to draft a pre-print of my PhD work, hopefully for publication. I am committed to improving my results and presentation, and Amaury’s paper certainly provides some inspiration is this direction.
As things stand I do not have to tools to develop results as good as Amaury’s. Therefore I am trying to develop my understanding of compact quantum groups and their representation theory. Afterwards I can hopefully study some of the remaining further problems mentioned in the thesis.
As suggested by Uwe Franz, representation theoretic methods (such as presented by Diaconis (1988) for the classical case), might be useful for analysing random walks on quantum homogeneous spaces.
J.P. McCarthy: Math Page 
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February 16, 2018 at 5:40 pm
Random Walks: Questions for The Future | J.P. McCarthy: Math Page
[…] at in my study of random walks on quantum groups. Following work of Amaury Freslon, a number of these questions have been answered. In exchange for solving these problems, Amaury has very kindly suggested some […]