Variational formula for the logarithmic potential of free additive convolutions
Abstract
We establish a general variational formula for the logarithmic potential of the free additive convolution of two compactly supported probability measure on . 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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