Critical dynamics of long range models on Dynamical L\'evy Lattices

We investigate critical equilibrium and out of equilibrium properties of a ferromagnetic Ising model in one and two dimension in the presence of long range interactions, $J_{ij}\propto r^{-(d+\sigma)}$. We implement a novel 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 $\sigma$. 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 $\sigma$, highlighting that the onset of short range regime for the dynamical critical properties appears to occur at a value of $\sigma$ which differs from the equilibrium one.