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We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limitwhere $\frac{N \beta}{2} \to c $, called the high temperature regime and showthat it can be used to construct a promising one-parameter interpolation, withparameter $c$ between the classical and the free convolution. This$c$-convolution has a simple interpretation in terms of another associatedfamily of distribution indexed by $c$, called the Markov-Krein transform: the$c$-convolution of two distributions corresponds to the classical convolutionof their Markov-Krein transforms. We derive first cumulants-moments relations,a central limit theorem, a Poisson limit theorem and shows several numericalexamples of $c$-convoluted distributions.

Rank one HCIZ at high temperature: interpolating between classical and free convolutions