Spontaneous repulsion in the $A+B\to0$ reaction on coupled networks

We study the transient dynamics of an $A+B \rightarrow 0$ process on a pair of randomly coupled networks, where reactants are initially separated. We find that, for sufficiently small fractions $q$ of cross-couplings, the concentration of $A$ (or $B$) particles decays linearly in a first stage and crosses over to a second linear decrease at a mixing time $t_x$. By numerical and analytical arguments, we show that for symmetric and homogeneous structures $t_x\propto(\nicefrac{\langle k \rangle}{q})\log(\nicefrac{\langle k \rangle}{q})$ where $\langle k \rangle$ is the mean degree of both networks. Being this behavior in marked contrast with a purely diffusive process---where the mixing time would go simply like $\langle k\rangle/q$---we identify the logarithmic slowing down in $t_x$ to be the result of a novel spontaneous mechanism of {\em repulsion} between the reactants $A$ and $B$ due to the interactions taking place at the networks' interface. We show numerically how this spontaneous repulsion effect depends on the topology of the underlying networks.