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 $\mathbb{R}$. 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.

2 figures, 25 pages