Variational formula for the logarithmic potential of free additive convolutions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912446142218240 |
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| author | Concetti, Francesco Belius, David Genovese, Giuseppe |
| author_facet | Concetti, Francesco Belius, David Genovese, Giuseppe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational formula for the logarithmic potential of free additive convolutions Concetti, Francesco Belius, David Genovese, Giuseppe Probability 46L54, 31A10, 60B20 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. |
| title | Variational formula for the logarithmic potential of free additive convolutions |
| topic | Probability 46L54, 31A10, 60B20 |
| url | https://arxiv.org/abs/2506.19064 |