Topological Dependence of Universal Correlations in Multi-Parameter Hamiltonians

Universality of correlation functions obtained in parametric random matrix theory is explored in a multi-parameter formalism, through the introduction of a diffusion matrix $D_{ij}(R)$, and compared to results from a multi-parameter chaotic model. We show that certain universal correlation functions in 1-d are no longer well defined by the metric distance between the points in parameter space, due to a global topological dependence on the path taken. By computing the density of diabolical points, which is found to increases quadratically with the dimension of the space, we find a universal measure of the density of diabolical points in chaotic systems.