In this thesis we study criticality in the context of the dissipative Abelian sandpile model. The model is linked to a simple trapped random walk, giving a practical method to determine criticality for certain landscapes of dissipative sites. The main results concern the lifetime of the random walk, especially the divergence of its first moment for traps placed on spherical shells. For the one dimensional case the point of divergence is determined with reasonable precision. In higher dimensions the divergence is shown to be possible for an infite amount of shells. The connection between the sandpile model and a random walk is shown mathematically and further researched via simulation.

In this thesis, we study stochastic duality under hydrodynamic scaling in the context of interacting particles on a grid. The approach is inspired and motivated by the relation between duality and local equilibria. We identify duality relations in terms of the expectation of the density field for which the hydrodynamic limit is recovered. This is initially done both for symmetric inclusion and exclusion processes as well as for independent random walkers. We continue with the independent case and generalize to particles that also possess a, possibly scale-dependent, internal energy state. The results in this context assume generator convergence under scaling and are illustrated using run-and-tumble systems. This work also includes examples concerning instances of run-and-tumble processes that do not have convergence on a generator level. Apart from run-and-tumble processes, we examine the effect of reservoirs on the relevant duality relations and macroscopic profiles. The reservoirs are found to correspond with boundary conditions for the macroscopic profile.