Critical dynamics of long-range models on dynamical L??vy lattices

We investigate critical equilibrium and out of equilibrium properties of a ferromagnetic Ising model in one and two dimensions in the presence of long-range interactions, Ji j ∝ r−(d+σ ). We implement a local dynamics on a dynamical Lévy lattice, that correctly reproduces the static critical exponents known in the literature, as a function of the interaction parameter σ. Due to its locality the algorithm can be applied to investigate dynamical properties, of both discrete and continuous long-range models. We consider the relaxation time at the critical temperature and we measure the dynamical exponent z as a function of the decay parameter σ, highlighting that the onset of the short-range regime for the dynamical critical properties appears to occur at a value of σ which differs from the equilibrium one.