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Journal Articles Japanese Journal of Mathematics Year : 2016

Free analysis and random matrices


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.
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ensl-01400950 , version 1 (22-11-2016)



Alice Guionnet. Free analysis and random matrices. Japanese Journal of Mathematics, 2016, 11 (1), pp.33-68. ⟨10.1007/s11537-016-1489-1⟩. ⟨ensl-01400950⟩
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