2016 Fiscal Year Research-status Report
Random Matrix Theory and its applications
| Project/Area Number |
26800048
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | Random Matrices |
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| Outline of Annual Research Achievements |
During the last year, I completed the following projects:
(1) together with Pierre Yves Gaudreau Lamarre (Princeton) we characterized star freeness for tensor products of unitaries. We proved that in many cases, tensoring by unitaires is the only option to sustain freeness. Our paper was published in Advances in Applied Mathematics.
(2) together with Termeh Kousha, Rafal Kulik, Tomasz Szarek and Karol Zyczkowski, we modified our paper “The accessibility of convex bodies and derandomization of the hit and run algorithm” and had it accepted in Journal of Convex Analysis.
(3) Together with Sho Matsumoto, we produced the preprint "Weingarten calculus via orthogonality relations: new applications". In this paper, we obtain optimal uniform asymptotics for the Weingarten function.
(4) Together with Konrad Szymanski, Tomasz Szare and Zyczkowski, we produced the preprint "Convex set of quantum states with positive partial transpose analysed by hit and run algorithm" that was accepted in Journal of Physics A. In this paper, we study the uniform measure on PPT states and obtain some partial results about its asymptotic distribution and its asymptotic norm. We also conjecture a limiting distribution for the eigenvalues.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
I solved many new problems that I had not expected I would consider nor solve during this year. However, I did not finish a joint work with Male and another joint work with Metcalfe. I also initiated discussions with Borodin (MIT) that I will continue to do in the future.
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| Strategy for Future Research Activity |
My plans to finish the project are as follows:
1- continue discussions with Alexei Borodin about fluctuation of random matrices arising from representation theory
2- continue my research project with Male (and include Bao too) on norm convergence of random matrices arising with tensor products.
3- continue to work with Nechita and Metcalfe on Riemann-Hilbert techniques in quantum information theory
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| Causes of Carryover |
1- Alexei Borodin (MIT): I knew I would meet with Borodin at IHP Paris in France in February 2017 and I wanted to discuss with him on some of my results and whether the point processes associated are determinantal or not. I did not have enough time to discuss with him in Paris and knew he was coming to Japan in April so I invited him to continue discussion on this project in Kyoto.
2- Camille Male (Bordeaux): we have a joint project (with him and Gaudreau Lamarre), we started two years ago and for which we needed meeting. However we could not meet earlier so he will come in May and June in Kyoto for collaboration.
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| Expenditure Plan for Carryover Budget |
My plan is parallel to the reason given above:
1- the first part is already conducted: I invited Alexei Borodin to visit me for one week in Kyoto in April. We completed our discussion. It will hopefully lead to joint research, we decided for now to continue the next round of investigations on
2- During the visit of Male in Kyoto, we will study the operator norm of polynomials of random matrices with a tensor stucture with the size of both tensors going to infinity. We plan to use the technology of traffics developed by Male in the last few years.
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| Remarks |
This website contains all information about my publications, CV and research activities
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