Japanese Journal of Mathematics

, Volume 11, Issue 1, pp 33–68 | Cite as

Free analysis and random matrices

Takagi Lectures


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.

Keywords and phrases

random matrices non-commutative measure Schwinger–Dyson equation 

Mathematics Subject Classification (2010)

15A52 (primary) 46L50 46L54 (secondary) 


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Copyright information

© The Mathematical Society of Japan and Springer Japan 2016

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Centre national de la recherche scientifiqueParis cedex 16France

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