Rank one HCIZ at high temperature: interpolating between classical and free convolutions – Archive ouverte HAL

Pierre Mergny ^{1, 2} Marc Potters ³

Pierre Mergny, Marc Potters. Rank one HCIZ at high temperature: interpolating between classical and free convolutions. SciPost Physics, SciPost Foundation, 2022, 12 (1), pp.022. ⟨10.21468/SciPostPhys.12.1.022⟩. ⟨hal-03738535⟩

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to c N β 2 → c , called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretation in terms of another associated family of distribution indexed by c, called the Markov-Krein transform: the c-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.

• 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
• 2. CFM – Capital Fund Management
• 3. CFM – Capital Fund Management