We calculate equilibrium solutions for Ising spin models on 'small world' lattices, which are constructed by super-imposing random and sparse Poissonian graphs with finite average connectivity c onto a one-dimensional ring. The nearest neighbour bonds along the ring are ferromagnetic, whereas those corresponding to the Poisonnian graph are allowed to be random. Our models thus generally contain quenched connectivity and bond disorder. Within the replica formalism, calculating the disorder-averaged free energy requires the diagonalization of replicated transfer matrices. In addition to developing the general replica symmetric theory, we derive phase diagrams and calculate effective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. 'small world' magnets), and that of ±J random sparse long-range bonds (i.e. 'small world' spin-glasses).

Abstract. We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs our macroscopic laws close already in terms of the singlespin path probabilities at zero external field. For the general case of arbitrary graph symmetry we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.

We consider a particle dragged through a medium at constant temperature as described by a Langevin equation with a time-dependent potential. The time dependence is specified by an external protocol. We give conditions on potential and protocol under which the fluctuations of the dissipative work satisfy an exact symmetry for all times. We also present counterexamples to that fluctuation theorem when our conditions are not satisfied. Finally, we consider the dissipated heat, which differs from the work by a temporal boundary term. We explain why there is a correction to the standard fluctuation theorem due to the unboundedness of that temporal boundary. However, the corrected fluctuation relation has again a general validity.

Abstract. We solve a class of attractor neural network models with a mixture of 1D nearestneighbour interactions and infinite-range interactions, which are both of a Hebbian-type form. Our solution is based on a combination of mean-field methods, transfer matrices, and 1D random-field techniques, and is obtained both for Boltzmann-type equilibrium (following sequential Glauber dynamics) and Peretto-type equilibrium (following parallel dynamics). Competition between the alignment forces mediated via short-range interactions, and those mediated via infinite-range ones, is found to generate novel phenomena, such as multiple locally stable 'pure' states, first-order transitions between recall states, 2-cycles and non-recall states, and domain formation leading to extremely long relaxation times. We test our results against numerical simulations and simple benchmark cases, and find excellent agreement.

We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so, we generalize existing versions of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally treelike interactions. We illustrate our results via application to, e.g., +/-J spin glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested against Monte Carlo simulations.