Free analysis and random matrices
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11537_2016_Article_1489.pdf
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1.02 MB
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Author(s)
Guionnet, Alice
Date Issued
May 2016
Journal
Japanese Journal of Mathematics
Publisher
Springer Japan
Citation
Guionnet, Alice. “Free Analysis and Random Matrices.” Japanese Journal of Mathematics 11, no. 1 (May 16, 2016): 33–68.
Version
Author's final manuscript
Abstract
We describe the Schwinger–Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes, we shall discuss when this equation uniquely defines the system and in such a case how it leads to deep properties of the solution. This analysis can be extended to systems which approximately satisfy these equations, such as random matrices or Coulomb gas interacting particle systems.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s11537-016-1489-1
