We test the property of ultrametricity for the spin-glass three-dimensional Edwards-Anderson model in zero magnetic field with numerical simulations up to ${20}^3$ spins. We find an excellent agreement with the prediction of the mean field theory. Since ultrametricity is not compatible with a trivial structure of the overlap distribution, our result contradicts the droplet theory.

The main goal of the paper is to prove central limit theorems for the magnetization rescaled by \(\sqrt{N}\) for the Ising model on random graphs with N vertices. Both random quenched and averaged quenched measures are considered. We work in the uniqueness regime \(\beta >\beta _c\) or \(\beta >0\) and \(B\ne 0\), where \(\beta \) is the inverse temperature, \(\beta _c\) is the critical inverse temperature and B is the external magnetic field. In the random quenched setting our results apply to general tree-like random graphs (as introduced by Dembo, Montanari and further studied by Dommers and the first and third author) and our proof follows that of Ellis in \(\mathbb {Z}^d\). For the averaged quenched setting, we specialize to two particular random graph models, namely the 2-regular configuration model and the configuration model with degrees 1 and 2. In these cases our proofs are based on explicit computations relying on the solution of the one dimensional Ising models.

The authors characterize the chaotic attractors of the Lorenz system associated with R=28 and R=60 in terms of the unstable periodic orbits and their eigenvalues. While the Hausdorff dimension is approximated with very good accuracy in both cases, the topological entropy is computed, in an exact sense only for R=28.

Usually, in a nonequilibrium setting, a current brings mass from the highest density regions to the lowest density ones. Although rare, the opposite phenomenon (known as “uphill diffusion”) has also been observed in multicomponent systems, where it appears as an artificial effect of the interaction among components. We show here that uphill diffusion can be a substantial effect, i.e., it may occur even in single component systems as a consequence of some external work. To this aim we consider the two-dimensional ferromagnetic Ising model in contact with two reservoirs that fix, at the left and the right boundaries, magnetizations of the same magnitude but of opposite signs.We provide numerical evidence that a class of nonequilibrium steady states exists in which, by tuning the reservoir magnetizations, the current in the system changes from “downhill” to “uphill”. Moreover, we also show that, in such nonequilibrium setup, the current vanishes when the reservoir magnetization attains a value approaching, in the large volume limit, the magnetization of the equilibrium dynamics, thus establishing a relation between equilibrium and nonequilibrium properties.