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arXiv · 2506.19064

Variational formula for the logarithmic potential of free additive convolutions

Francesco Concetti·David Belius·Giuseppe Genovese
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Abstract

We establish a general variational formula for the logarithmic potential of the free additive convolution of two compactly supported probability measure on $\R$. The formula is given in terms of the $R$-transform of the first measure, and the logarithmic potential of second measure. The result applies in particular to the additive convolution with the semicircle or Marchenko-Pastur laws, for which the formula simplifies. The logarithmic potential of additive convolutions appears for instance in estimates of the determinant of sums of independent random matrices.

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Francesco Concetti, David Belius, Giuseppe Genovese. 2025-06-23. Variational formula for the logarithmic potential of free additive convolutions. https://arxiv.org/abs/2506.19064

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