Arbitrary unitarily invariant random matrix ensembles and supersymmetry

Abstract

We generalize the supersymmetry method in Random Matrix Theory to arbitrary rotation invariant ensembles. Our exact approach further extends a previous contribution in which we constructed a supersymmetric representation for the class of norm-dependent Random Matrix Ensembles. Here, we derive a supersymmetric formulation under very general circumstances. A projector is identified that provides the mapping of the probability density from ordinary to superspace. Furthermore, it is demonstrated that setting up the theory in Fourier superspace has considerable advantages. General and exact expressions for the correlation functions are given. We also show how the use of hyperbolic symmetry can be circumvented in the present context in which the non-linear σ model is not used. We construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the Green functions.

The Correlated Jacobi and the Correlated Cauchy–Lorentz Ensembles

“…We also derived a supersymmetric integral for the k-point generating function of the correlated Jacobi ensemble and the correlated Cauchy-Lorentz ensemble via the projection formula [10,11] in combination with generalized Hubbard-Stratonovich transformation [42,44,45] and the superbosonization formula [40][41][42]. The resulting integral over the supermatrices looks similar to the one of the correlated Wishart ensemble [26].…”

“…Due to the invariance of the integrand in Eq. (77) under A → U A for an arbitrary U ∈ O(p) for β = 1 and U ∈ U(p) for β = 2, the integrand only depends on the invariants tr AA † m for m ∈ N. These invariants are equal to the superinvariants tr AA † m = str A † A m , see [38,39,44,45]. Employing this duality in the generating function (77), we arrive at…”

“…Alternative to the superbosonization formula one can also choose the generalized HubbardStratonovich transformation [42,44,45], see third equality of Eq. (19), which we employ in section 5.…”

“…The first can be understood as the contour representation of the latter which is some kind of a high dimensional residue theorem. The distribution I n 2 (σ) in the generalized Hubbard-Stratonovich transformation is the supersymmetric Ingham-Siegel integral [44,45],…”

“…The dimension of the supermatrix model is (2|2) × (2|2) and thus twice as large as for the kernel of the complex matrices. We apply the generalized Hubbard-Stratonovich transformation [42,44,45] for k = 1, third equality of Eq. (19), which explicitly reads in this case…”