Random field Ising chain and neutral networks with synchronous dynamics

We first present an exact solution of the one-dimensional random-field Ising model in which spin-updates are made fully synchronously, i.e. in parallel (in contrast to the more conventional Glauber-type sequential rules). We find transitions where the support of local observables turns from a continuous interval into a Cantor set and we show that synchronous and sequential random-field models lead asymptotically to the same physical states. We then proceed to an application of these techniques to recurrent neural networks where 1D short-range interactions are combined with infinite-range ones. Due to the competing interactions these models exhibit phase diagrams with first-order transitions and regions with multiple locally stable solutions for the macroscopic order parameters