2009/08/31 by S. Mandt, S Mandt and M R Zirnbauer, M. R. Zirnbauer
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Circular ensemble #Combinatorics #Eigenvalues and eigenvectors #Fourier transform #Free probability #Hermitian matrix #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Random variable #Statistics #Supersymmetry #Unitary state #Universality (dynamical systems) #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8113/43/2/025201
published as J. Phys. A 43 (2010) 025201 · 38 pages, 3 figures, published version, minor changes in Section 6
openalex publication_date 2009/12/10 · arxiv created 2010/01/05 · arxiv updated 2010/01/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by 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.