Universality of Correlations of Levels¶with Discrete Statistics

Abstract:

We study the statistics of a system of N random levels with integer values, in the presence of a logarithmic repulsive potential of Dyson type. This problem arises in sums over representations (Young tableaux) of GL(N) in various matrix problems and in the study of statistics of partitions for the permutation group. The model is generalized to include an external source and its correlators are found in closed form for any N. We reproduce the density of levels in the large N and double scaling limits and the universal correlation functions in Dyson's short-distance scaling limit. We also study the statistics of small levels.