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Free Probability, Random Matrices and Disorder – CME 510

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Published on Apr 17, 2015

Free probability, random matrices and disorder in organic semiconductors
Jiahao Chen, MIT CSAIL, jiahao@mit.edu

Random matrix theory has long been used to study the spectral
properties of physical systems, and has led to a rich interplay
between probability theory and physics [1]. Historically, random
matrices have been used to model physical systems with random
fluctuations, or systems whose eigenproblems were too difficult to
solve numerically. This talk explores applications of RMT to the
physics of disorder in organic semiconductors [2,3]. Revisiting the
old problem of Anderson localization [4] has shed new light on the
emerging field of free probability theory [5]. I will discuss the
implications of free probabilistic ideas for finite-dimensional random
matrices [6], as well as some hypotheses about eigenvector locality.
Algorithms are available in the RandomMatrices.jl package [7] written
for the Julia programming language.

[1] M. L. Mehta. Random matrices, 3/e, Academic Press, 2000.
[2] J. Chen, E. Hontz, J. Moix, M. Welborn, T. Van Voorhis, A. Suarez, R. Movassagh, and A. Edelman. Error analysis of free probability approximations to the density of states of disordered systems. Phys. Rev. Lett. (2012) 109:36403.
[3] M. Welborn, J. Chen, and T. Van Voorhis. Densities of states for disordered systems from free probability. Phys. Rev. B (2013) 88:205113.
[4] P. W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev. (1958) 109:1492--1505.
[5] D. Voiculescu. Addition of certain non-commuting random variables. J. Functional Anal. (1986) 66:323--346.
[6] J. Chen, T. Van Voorhis, and A. Edelman. Partial freeness of random matrices. arXiv:1204.2257
[7] https://github.com/jiahao/RandomMatri...

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