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In the computation of the TAP complexity, originally carried out by Bray and
Moore, a fundamental step is to calculate the determinant of a random Hessian.
As the replica method does not give a clear prescription, physicists debated
how to perform this computation and its consequences on the TAP complexity for
a long time. In this paper we prove the original Bray and Moore formula for the
behaviour of the determinant at exponential scale to be correct, and compute an
important prefactor coming from a small outlier in the spectrum.
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