Cutoff Thermalization for Ornstein–Uhlenbeck Systems with Small Lévy Noise in the Wasserstein Distance

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems $$(X^\varepsilon _t(x))_{t\geqslant 0}$$ ( X t ε ( x ) ) t ⩾ 0 with $$\varepsilon $$ ε -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, $$\alpha $$ α -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp $$\infty /0$$ ∞ / 0 -collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $$\mu ^\varepsilon $$ μ ε along a time window centered on a precise $$\varepsilon $$ ε -dependent time scale $$\mathfrak {t}_\varepsilon $$ t ε . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result $$\mathcal {W}_p(X^\varepsilon _{t_\varepsilon + r}(x), \mu ^\varepsilon ) \cdot \varepsilon ^{-1} \rightarrow K\cdot e^{-q r}$$ W p ( X t ε + r ε ( x ) , μ ε ) · ε - 1 → K · e - q r for any $$r\in \mathbb {R}$$ r ∈ R as $$\varepsilon \rightarrow 0$$ ε → 0 for some spectral constants $$K, q>0$$ K , q > 0 and any $$p\geqslant 1$$ p ⩾ 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of $$\mathcal {Q}$$ Q . Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $$\varepsilon $$ ε -small Brownian motion or $$\alpha $$ α -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

Mixing times and cutoffs in open quadratic fermionic systems

In classical probability theory, the term cutoff describes the property of some Markov chains to jump from (close to) their initial configuration to (close to) completely mixed in a very narrow window of time. We investigate how coherent quantum evolution affects the mixing properties in two fermionic quantum models (the ``gain/loss'' and ``topological'' models), whose time evolution is governed by a Lindblad equation quadratic in fermionic operators, allowing for a straightforward exact solution. We check that the cutoff phenomenon extends to the quantum case and examine how the mixing properties depend on the initial state. In the topological case, we further show how the mixing properties are affected by the presence of a long-lived edge zero mode when taking open boundary conditions.

A NOTE ON THE ASYMPTOTIC BEHAVIOR OF THE HEIGHT FOR A BIRTH-AND-DEATH PROCESS

Recently, the asymptotic mean value of the height for a birth-and-death process is given in Videla [Videla, L.A. (2020)]. We consider the asymptotic variance of the height in the case when the number of states tends to infinity. Further, we prove that the heights exhibit a cutoff phenomenon and that the normalized height converges to a degenerate distribution.

Dynamics of interacting particle systems

We collect here recent results covering various aspects of the dynamical properties of interacting particle systems. In Section 1 we study the hydrodynamic limit of a facilitated exclusion process. Section 2 evidences a cutoff phenomenon for the mixing time of the weakly asymmetric exclusion process. Section 3 presents a study of the infection time in the Duarte model. Finally, Section 4 presents the study of a front propagation in the FA-If model.