Let $\mu$ and $\nu$ be compactly supported probability measures on the real line with densities with respect to Lebesgue measure. We show that for all large real $z$, if $\mu \boxplus \nu$ is their additive free convolution, we have

$$\begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) = \sup_{\Pi} \left\{ \mathbb{E}_\Pi[\log(z - (X+Y)] - \mathcal{E}[\Pi]+\mathcal{E}[\mu]+\mathcal{E}[\nu] \right\}, \end{equation*}$$ where the supremum is taken over all probability laws $\Pi$ on $\mathbb{R}^2$ for a pair of real-valued random variables $(X,Y)$ with respective marginal laws $\mu$ and $\nu$, and given a probability law $P$ with density function $f$ on $\mathbb{R}^k$, $\mathcal{E}[P] := \int_{\mathbb{R}^k} f \log f$ is its classical entropy. We prove similar formulas for the multiplicative free convolution $\mu \boxtimes \nu$ and the free compression $[\mu]_\tau$ of probability laws. We use our formulation to derive several new inequalities relating free and classical convolutions of random variables, such as $$\begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) \geq \mathbb{E}[\log(z - (X+Y)], \end{equation*}$$ valid for all large $z$, where on the right-hand side $X,Y$ are independent classical random variables with respective laws $\mu,\nu$. Our approach is based on applying a large deviation principle on the symmetric group to the celebrated quadrature formulas of Marcus, Spielman and Srivastava. This talk is based in joint work with Samuel G. G. Johnston.

Bio: Octavio Arizmendi is a Mexican mathematician. He finished his PhD in 2012 at the University of Saarbrucken under the supervision of Roland Speicher. He then went to CIMAT as a postdoc in 2013, where he since later became a professor. Octavio works in Free Probability and more generally Non-Commutative Probability, where he has made contribution in various aspects of the theory, specially in the combinatorial side.