For compactly supported probability measures, the logarithmic potential of their free additive convolution equals the minimum of an explicit variational energy, with explicit forms for semicircle and Marchenko-Pastur convolutions.
On the determinant in Bray-Moore's TAP complexity formula
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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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Variational formula for the logarithmic potential of free additive convolutions
For compactly supported probability measures, the logarithmic potential of their free additive convolution equals the minimum of an explicit variational energy, with explicit forms for semicircle and Marchenko-Pastur convolutions.