Critical dynamics of long-range models on dynamical Lévy 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, ${J}_ij\ensuremath{\propto}{r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$. We implement a local dynamics on a dynamical L\'evy lattice, that correctly reproduces the static critical exponents known in the literature, as a function of the interaction parameter $\ensuremath{\sigma}$. 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 $\ensuremath{\sigma}$, highlighting that the onset of the short-range regime for the dynamical critical properties appears to occur at a value of $\ensuremath{\sigma}$ which differs from the equilibrium one.