We study the long-range one-dimensional Riesz gas on the circle, a continuous system of particles interacting through a Riesz (i.e inverse power) kernel. We establish nearoptimal rigidity estimates on gaps valid at any scale. Leveraging on these local laws and using a Stein method, we provide a quantitative Central Limit Theorem for linear statistics. The proof is based on a mean-field transport and on a fine analysis of the fluctuations of local error terms through the study of Helffer-Sjöstrand equations. The method can handle very singular test-functions, including characteristic functions of intervals, using a comparison principle for the Helffer-Sjöstrand equation.
We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, entropy and Wasserstein. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions. Contents 1. Introduction and main results 1 2. Additional comments and open problems 9 3. Cutoff phenomenon for the OU 14 4. General exactly solvable aspects 15 5. The random matrix cases 18 6. Cutoff phenomenon for the DOU in TV and Hellinger 21 7. Cutoff phenomenon for the DOU in Wasserstein 26 Appendix A. Distances and divergences 27 Appendix B. Convexity and its dynamical consequences 30 Acknowledgements 33 References 33
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