![Interpolation between random matrices and free operators, and application to Quantum Information Theory](/media/cache/video_light/uploads/video/2024-07-08_parraud.mp4-114227d90047b2656ca9c1be23581a6c-video-affed710d41fa907b23c2824d01b028a.jpg)
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Interpolation between random matrices and free operators, and application to Quantum Information Theory
By Félix Parraud
![The mean-field limit of non-exchangeable integrate and fire systems](/media/cache/video_light/uploads/video/2024-06-27_Jabin_1.mp4-791d3866c3e8b85d50079528cc15f5ec-video-6df69a073a7fc8593faeb3d1da32c5ff.jpg)
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The mean-field limit of non-exchangeable integrate and fire systems
By Pierre-Emmanuel Jabin
Appears in collection : Modern Analysis Related to Root Systems with Applications / Analyse moderne liée aux systèmes de racines avec applications
We study Bessel and Dunkl processes $\left(X_{t, k}\right)_{t>0}$ on $\mathbb{R}^{N}$ with possibly multivariate coupling constants $k \geq 0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_{N}$ these Bessel processes are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types $\mathrm{A}$ and $\mathrm{B}$ and derive analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N \rightarrow \infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type $\mathrm{B}$ new non-symmetric semicircle-type limit distributions on $\mathbb{R}$ appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko-Pastur limit laws for the empirical measures of the zeroes of Hermite and Laguerre polynomials respectively. (based on joint work with Michael Voit)