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Relaxation to equilibrium of generalized East processes on ${\mathbb{Z}}^d$: Renormalization group analysis and energy-entropy competition

We consider a class of kinetically constrained interacting particle systems on ${\mathbb{Z}}^d$ which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of ${\mathbb{Z}}^d$, to each occupation variable $η_x\in\{0,1\}$ a new value is proposed by tossing a $(1-q)$-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For $d=1$, the constraint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case, the process is the well-known East process. On ${\mathbb{Z}}^2$, the West or the South neighbor of the updating vertex must contain a vacancy, similarly, in higher dimensions. Despite of their apparent simplicity, in the limit $q\searrow0$ of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on ${\mathbb{Z}}^d$ scales as the $1/d$-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next, we compute the relaxation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multiscale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally, we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex $x$ out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the $\ell_1$-norm of $x$ but also on its direction.