Variational formula for the logarithmic potential of free additive convolutions (2506.19064v1)

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.

• The paper presents a new variational formula for the logarithmic potential of free additive convolution, simplifying analysis for semicircle and Marchenko-Pastur laws.
• The authors leverage the R-transform and inverse function theorem to derive practical estimates for determinants of sums of independent random matrices.
• The results provide theoretical insights with implications for random matrix theory, high-dimensional landscapes, and machine learning applications.

Variational Formula for the Logarithmic Potential of Free Additive Convolutions

This paper presents a comprehensive paper on the logarithmic potential associated with the free additive convolution of two compactly supported probability measures on the real line. The authors, David Belius, Francesco Concetti, and Giuseppe Genovese, have established a variational formula for this potential, expressed in terms of the $R$-transform of the first measure and the logarithmic potential of the second measure. Their results hold significant implications across a range of applications, notably including estimates for the determinant of sums of independent random matrices, which find use in fields like high-dimensional random landscapes and machine learning theory.

Theoretical Framework and Key Results

The paper begins by defining the logarithmic potential of a probability measure $\mu$ as $U_\mu(z) := \int \log|z - x| \mu(dx),$ which is an instrumental concept in potential theory. The authors focus on the scenario where two probability measures, $\mu$ and $\nu$, both have compact supports. The crux of their inquiry involves deriving the expression for the logarithmic potential $U_{\mu \boxplus \nu}$ of the free additive convolution $\mu \boxplus \nu$.

A cornerstone of this work is the derivation of a new variational formula for the logarithmic potential of the free additive convolution, tailored for cases when $\mu$ conforms to particular laws such as the semicircle or Marchenko-Pastur. In such cases, the formula can be notably simplified, allowing for enhanced computational feasibility.

Semicircle Law and Marchenko-Pastur Law

For a semicircle law $\mu = \musc[\beta]$ and a compactly supported measure $\nu$, the variational formula simplifies to: $U_{\mu\boxplus\nu}(z) = \inf_{g \in (0,g^*_{\mu \boxplus \nu}) \, \text{such that}\, \beta^2 g < \supp_{-}\nu - z} \left\{ \frac{\beta^2 g^2}{2} + \int\log( \lambda - z - \beta^2 g) \nu(d\lambda) \right\}.$

For the Marchenko-Pastur law $\mu$, the formula is similarly articulated, capturing context-specific dynamics through adjusted parameters and integrals aligned with the associated probability distributions' characteristics.

Methodology and Analytical Strategy

The method relies heavily on manipulating the structure of the $R$-transform and its associated properties. The $R$-transform function's derivative plays a key role in evaluating critical points of the variational expression. The authors employ classical principles from real analysis and potential theory to underpin their variational approach, while extensive use of the inverse function theorem affirms the theoretical underpinnings of their deductions.

Implications and Future Work

The implications of these findings are vast, especially in areas necessitating the understanding of random matrix eigenvalues' distribution, such as spin glasses or in the analysis of complexity landscapes in neural networks. The authors posit these results could bolster efforts in studying phase transitions in complex systems and improve estimations related to expected numbers of critical points in energy landscapes.

Furthermore, the potential for extending these results to more general frame structures opens intriguing avenues for further investigation. Given the foundational nature of the insight presented, the paper raises several avenues for subsequent empirical and theoretical work, particularly in verifying these variational formulas' utility in practical scenarios across applied probability and statistical mechanics.

In conclusion, this paper enriches the mathematical framework surrounding free probabilities, offering robust formulas that extend our abilities to compute and analyze properties of convoluted distributions. This work will likely serve as a touchstone for future examinations into the intricate relationships facilitated by free additive convolutions.