Some remarks on the attractor behaviour in ELKO cosmology

Recent results on the dynamical stability of a system involving the interaction of the ELKO spinor field with standard matter in the universe have been reanalysed, and the conclusion is that such system does not exhibit isolated stable points that could alleviate the cosmic coincidence problem. When a constant parameter $\delta$ related to the potential of the ELKO field is introduced in the system however, stable fixed points are found for some specific types of interaction between the ELKO field and matter. Although the parameter $\delta$ is related to an unknown potential, in order to satisfy the stability conditions and also that the fixed points are real, the range of the constant parameter $\delta$ can be constrained for the present time and the coincidence problem can be alleviated for some specific interactions. Such restriction on the ELKO potential opens possibility to apply the ELKO field as a candidate to dark energy in the universe, and so explain the present phase of acceleration of the universe through the decay of the ELKO field into matter.Comment: 17 pages, section III with minor changes and section IV rewritten with a new analysi

Universality class for bootstrap percolation with m=3m=3 on the cubic lattice

We study the $m=3$ bootstrap percolation model on a cubic lattice, using Monte Carlo simulation and finite-size scaling techniques. In bootstrap percolation, sites on a lattice are considered occupied (present) or vacant (absent) with probability $p$ or $1-p$, respectively. Occupied sites with less than $m$ occupied first-neighbours are then rendered unoccupied; this culling process is repeated until a stable configuration is reached. We evaluate the percolation critical probability, $p_c$, and both scaling powers, $y_p$ and $y_h$, and, contrarily to previous calculations, our results indicate that the model belongs to the same universality class as usual percolation (i.e., $m=0$). The critical spanning probability, $R(p_c)$, is also numerically studied, for systems with linear sizes ranging from L=32 up to L=480: the value we found, $R(p_c)=0.270 \pm 0.005$, is the same as for usual percolation with free boundary conditions.Comment: 11 pages; 4 figures; to appear in Int. J. Mod. Phys.