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Random Matrices

Random matrices were first used in the 50s as a modelization of Hamiltonians of complex nuclear systems, with the hope that statistical properties of energy levels are well described by the statistics of eigenvalues of random matrices. These ideas were then successfully applied to other areas of physics, including quantum chaos and disordered systems, but also to number theory (statistics of the zeroes of the Riemann zeta function).

Another, separate, application of random matrices appeared in the late 70s when they were used as toy models for the large N limit of SU(N)-invariant gauge theories : these "matrix models" are zero-dimensional field-theories with well-understood Feynman diagram expansion, which can be viewed as a summation over discretized random surfaces. In the 80s this was used to provide a discretized version of two-dimensional quantum gravity. In the following years more mathematical applications appeared in which matrix models allowed for the exact calculation of certain generating series of enumerative combinatorics (planar maps).

The research team on random matrices is comprised of P. Zinn-Justin and J.-B. Zuber. In the recent years they have worked on applying matrix model techniques to enumeration problems in knot theory. They have also collaborated with mathematicians within the framework of the ANR program : GranMA.