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Of this equation can be rewritten as Δ 1 ∂i (Δ2 ∂i ξ). Δ2 (13) This results in the diﬀusion equation 1 ∂t F = ∂i2 F. 2 (14) The solution of this diﬀusion equation is given by F (X, t) = c 1 1 tn/2 dY Δ(y)e− 2t i (xi −yi )2 η(Y ), (15) with c an arbitrary constant. This result should be valid for any invariant function of the eigenvalues. An invariant function is also a symmetric function of the eigenvalues. The exponent in this equation can be factorized in symmetric functions of the integration variables and exp( i xi yi /t).
Phys. 32 (1983) 53. M. A. R. Zirnbauer, Phys. Rep. 129, 367 (1985). J. Wegner, Z. Phys. B49 (1983) 297.  V. Rittenberg and M. Scheunert, J. Math. Phys. 19 (1978) 709. R. Zirnbauer, Nucl. Phys. B 265 [FS15] (1968) 375. B. Gossiaux, Z. A. Weidenm¨ uller, coond-mat/9803362. R. M. Haldane, Phys. Rev. B52 (1995) 8729. 7 9 The Supersymmetric Method of Random Matrix Theory: The One-Point Function In previous lecture we have seen that the orthogonal polynomials method is a powerful method which has been applied successfully to many problems in Random Matrix Theory.
Let us perform the integral. After doing the Grassmann integrations we find DQF (Q) = 1 πi dadb ∂a F00 + ∂b F00 . a−b (21) This integral is not well defined if a and b are along the same path in the complex plane. For definiteness let us take a along the real axis and b along the imaginary axis. In polar coordinates a = ρ cos φ, (22) b = iρ sin φ. (23) Then ∂a F00 + ∂b F00 = ( a−b a−b ∂φ )F00 . ∂ρ + ρ iρ2 (24) The integral over φ of the second term gives zero provided that the point ρ = 0 is suitably regularized.