Zooming in on Local Level Statistics by Supersymmetric Extension of Free Probability

— We consider Hermitian N ×N random matrices H distributed according to a probability density dμN(H) = e−N Tr V (H)dH with analytic and uniformly convex V . From work by Zinn-Justin, Collins, and Guionnet & Maida it is known that the large-N limit of the scaled logarithm N−1 lnΩ(NK) of the Fourier transform Ω(K) of dμN(H) for K of finite rank, is a linear functional of the Voiculescu R-transform – a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.