By Pierre de la Harpe

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Phys. 32 (1991) 336. 3 10 The Supersymmetric Method of Random Matrix Theory: The Two-Point Function In this chapter we discuss the two-point correlation function for the GUE using the supersymmetric method. The eﬀective partition function is constructed by mainly relying on the symmetries of the generating function. For the calculation of the two-point function we refer to an article by Martin Zirnbauer and myself [6] and a very detailed article by Joe Zuk [7] that is based on a papers by Efetov [1] and Zirnbauer [3].

Notice that eq. (35) is singular for λ equal to one of the eigenvalues of b. 2. Show that the eigenvalues of a (1|1) super-matrix are given by σρ , a−b σρ = b− . a−b λ1 = a − (37) λ2 (38) Now let us consider a more complicated integral involving an invariant function of Q† Q with Q the (1|1) supermatrix given in eq (18). The matrix Qd aggerQ is then given by a∗ a + ρ∗ ρ a∗ σ + bρ∗ ∗ ∗ ∗ −aσ + b ρ b b − sigma∗ sigma Qd aggerQ = . (39) The eigenvalues of this matrix are given by λ1 = a∗ a + ρ∗ ρ − a∗ aσ ∗ σ + b∗ bρ∗ ρ + a∗ b∗ σρ + abσ ∗ ρ∗ (a∗ a + b∗ b)σσ + ρ∗ ρ , + a∗ a − b∗ b (a∗ a − b∗ b)2 λ2 = λ1 − a∗ a + b∗ b − σ ∗ σ − ρ∗ ρ.

By expression the trace in components one can easily show that ξ satisﬁes the diﬀusion equation 1 ∂t ξ = ∇2A ξ, 2 (4) ξ(A, 0) = η(A). (5) with initial condition The matrices A and B can be diagonalized by a unitary transformation A = U XU −1 , B = V Y V −1 . (6) The nonzero matrix elements of the diagonal matrices X and Y will be denoted by xi and yi , respectively. The Jacobian from the A variables to the U and X variables is given by dA = Δ2 (X)dXdΩA . (7) where the angular degrees of freedom arec contained in ΩA .