X-type and Y-type junction stability in domain wall networks

We develop an analytic formalism that allows one to quantify the stability properties of X-type and Y-type junctions in domain wall networks in two dimensions. A similar approach might be applicable to more general defect systems involving junctions that appear in a range of physical situations, for example, in the context of F- and D-type strings in string theory. We apply this formalism to a particular field theory, Carter's pentavac model, where the strength of the symmetry breaking is governed by the parameter $|\epsilon|< 1$. We find that for low values of the symmetry breaking parameter X-type junctions will be stable, whereas for higher values an X-type junction will separate into two Y-type junctions. The critical angle separating the two regimes is given by \alpha_c = 293^{\circ}\sqrt{|\epsilon|}$and this is confirmed using simple numerical experiments. We go on to simulate the pentavac model from random initial conditions and we find that the dominant junction is of \ytype for$|\epsilon| \geq 0.02$and is of \xtype for$|\epsilon| \leq 0.02$. We also find that for small$\epsilon$the evolution of the number of domain walls$\qsubrm{N}{dw}$in Minkowski space does not follow the standard$\propto t^{-1}$scaling law with the deviation from the standard lore being more pronounced as$\epsilon$is decreased. The presence of dissipation appears to restore the$t^{-1}$ lore.