Colloquium on May 18, 2009

Jean Zinn-Justin
CEA Saclay

Renormalization group approach to matrix models

The study of the statistical properties of random matrices of large size has a long history, beginning with Dyson's study of Gaussian ensembles and the discovery of the famous semi-circle law, with the goal to describe the statistical properties of the spectrum of complex Hamiltonians. Later, 't Hooft observed that in SU(N) non-Abelian gauge theories, to Feynman diagrams can be associated tessalated surfaces and that the large N expansion corresponds to an expansion in successive topologies. Using the idea of associating Feynman graphs to surfaces, some times later, it was realized that some ensembles of random matrices could be seen as toy models for quantum gravity called 2D quantum gravity, in the so-called double scaling limit. This has resulted in a tremendous expansion of random matrix theory, tackled with increasingly sophisticated mathematical methods and number of matrix models have been solved exactly. However, the somewhat paradoxical situation is that either models can be solved exactly or very little can be said. Since the solved models display critical points and univeral properties, it is tempting to use renormalization group ideas to reproduce universal properties, without solving models explicitly. Some attempt†along these lines is presented here. It is hoped that†some generalization of such methods could be applied to unsolved models.