Hydrodynamic Limit for the d-Facilitated Exclusion Process

Consider a periodic one-dimensional exclusion process with the dynamical constraint in which the particle at site x is prevented from jumping to $$x+1$$ x + 1 (or $$x-1$$ x - 1 ) unless the sites $$x-1,x-2,\ldots ,x-d+1$$ x - 1 , x - 2 , … , x - d + 1 (or $$x+1,x+2,\ldots ,x+d-1$$ x + 1 , x + 2 , … , x + d - 1 ) are all occupied and the site $$x+1$$ x + 1 (or $$x-1$$ x - 1 ) is empty. The case $$d=2$$ d = 2 was introduced by Basu et al. (Phys Rev E, 2009) and further studied by Blondel et al. (Ann Inst Henri Poincaré Probab Stat, 2020). Provided that the initial profile is suitably smooth and uniformly larger than the critical density $$\frac{d-1}{d}$$ d - 1 d , we prove the macroscopic density profile evolves under the diffusive time scaling according to a fast diffusion equation. This equation can be converted to the same equation as in Blondel et al. with initial profile greater than $$\frac{1}{d}$$ 1 d . The main ingredients in this proof are to verify properties of invariant measures like exponential decay of correlations and equivalence of ensembles. The difficulties arising from the constraint number d ( $$d>2$$ d > 2 ) are overcome by more delicate analysis.