Publication : t00/036

Abstract:
We solve the puzzle raised by Br\'ezin and Deo for random $N\times N$ matrices with a disconnected eigenvalues support: their calculation by orthogonal polynomials disagrees with previous mean field calculations. We show that this difference does not stem from a $\mathbb{Z}_2$ symmetry breaking, but from the discretizeness of the number of eigenvalues. This leads to additional terms (quasiperiodic in $N$) which must be added to the naive mean field expression. Our result invalidates the existence of a smooth topological large $N$ expansion and postulated universality properties of correlators. We derive for the general 2-cut case the expressions for the 2-point correlators and for the orthogonal polynomials in the large $N$ limit, and extend our results to any number of cuts and to non-real potentials.

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