Fig. 6.

A snapshot of ${\text{Φ}}_{n_0}(\mathit{n},t)$ at time t = 100 obtained from Eq. (5) with standard numerical methods [71,72]. Propagator with different configurations of defects, corresponding with Fig. 1, at t = 100. For all panels, the homogeneous propagator, ${\tilde{\phi }}_{n_0}(\mathit{n},z)$ is the 2D propagator with reflecting boundaries given in Eq. (23) of Ref. [68] (see also Appendix H). The parameters used are: a domain of size N = (10, 10), a localized initial condition with n₀ = (6, 6). We a use diffusion parameter of value q = (0.2, 0.2), which gives the following transition probabilities: in the bulk of the homogeneous lattice the probability of jumping to one of the four neighbors is A_r,s = 0.05 (with r ≠ s), while the probability of staying at the same site is A_r,r = 0.8. The reflecting barriers and other heterogeneities are super imposed on top of the probability. For panels (a) and (b), λ_v,u = A_v,u and λ_u,v = A_v,u (for all {u, v}∈ S) yielding perfectly reflecting barriers. For panel (c) ${\lambda }_{\mathit{\text{v}},\mathit{\text{u}}}=-\frac{1}{2}{\underset{\_}{\mathit{A}}}_{\mathit{\text{u}},\mathit{\text{u}}}$, and ${\lambda }_{\mathit{\text{u}},\mathit{\text{v}}}=-\frac{1}{2}{\underset{\_}{\mathit{A}}}_{\mathit{\text{v}},\mathit{\text{v}}}$. With this perturbation, when on one of the defective sites, the probability of staying is reduced to A_u,u = A_v,v = 0.4, while the probability of jumping to the non-neighbor is increased (from zero) to A_u,v = A_v,u = 0.4. Lastly, for panel (d), for each of the sticky sites w with k neighbors r₁,…, r_k we use ${\lambda }_{r_i,w}=\frac{1}{4}{\underset{\_}{\mathit{A}}}_{r_i,w}$ (see Sec. II A 3). For convenience we have omitted color bars for each panel as we are interested only in the relative differences of the occupation probability.