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Article Dans Une Revue Probability Theory and Related Fields Année : 2023

Universal cutoff for Dyson Ornstein Uhlenbeck process

Résumé

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.

Dates et versions

hal-03311688 , version 1 (02-08-2021)

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Citer

Jeanne Boursier, Djalil Chafaï, Cyril Labbé. Universal cutoff for Dyson Ornstein Uhlenbeck process. Probability Theory and Related Fields, 2023, 185 (1-2), pp.449-512. ⟨10.1007/s00440-022-01158-5⟩. ⟨hal-03311688⟩
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