Long-range epidemic spreading in a random environment

Emergence of disconnected clusters in heterogeneous complex systems

Percolation theory dictates an intuitive picture depicting correlated regions in complex systems as densely connected clusters. While this picture might be adequate at small scales and apart from criticality, we show that highly correlated sites in complex systems can be inherently disconnected. This finding indicates a counter-intuitive organization of dynamical correlations, where functional similarity decouples from physical connectivity. We illustrate the phenomenon on the example of the disordered contact process (DCP) of infection spreading in heterogeneous systems. We apply numerical simulations and an asymptotically exact renormalization group technique (SDRG) in 1, 2 and 3 dimensional systems as well as in two-dimensional lattices with long-ranged interactions. We conclude that the critical dynamics is well captured by mostly one, highly correlated, but spatially disconnected cluster. Our findings indicate that at criticality the relevant, simultaneously infected sites typically do not directly interact with each other. Due to the similarity of the SDRG equations, our results hold also for the critical behavior of the disordered quantum Ising model, leading to quantum correlated, yet spatially disconnected, magnetic domains.

Measuring neuronal avalanches in disordered systems with absorbing states

Power-law-shaped avalanche-size distributions are widely used to probe for critical behavior in many different systems, particularly in neural networks. The definition of avalanche is ambiguous. Usually, theoretical avalanches are defined as the activity between a stimulus and the relaxation to an inactive absorbing state. On the other hand, experimental neuronal avalanches are defined by the activity between consecutive silent states. We claim that the latter definition may be extended to some theoretical models to characterize their power-law avalanches and critical behavior. We study a system in which the separation of driving and relaxation time scales emerges from its structure. We apply both definitions of avalanche to our model. Both yield power-law-distributed avalanches that scale with system size in the critical point as expected. Nevertheless, we find restricted power-law-distributed avalanches outside of the critical region within the experimental procedure, which is not expected by the standard theoretical definition. We remark that these results are dependent on the model details.

Phase transitions of the random-bond Potts chain with long-range interactions

We study phase transitions of the ferromagnetic q-state Potts chain with random nearest-neighbor couplings having a variance Δ^{2} and with homogeneous long-range interactions, which decay with distance as a power r^{-(1+σ)}, σ>0. In the large-q limit the free-energy of random samples of length L≤2048 is calculated exactly by a combinatorial optimization algorithm. The phase transition stays first order for σ<σ_{c}(Δ)≤0.5, while the correlation length becomes divergent at the transition point for σ_{c}(Δ)<σ<1. In the latter regime the average magnetization is continuous for small enough Δ, but for larger Δ-according to the numerical results-it becomes discontinuous at the transition point, thus the phase transition is expected of mixed order.