Each chapter of this thesis is a self-contained work, studying the long-time ormany-particle limit of a different automaton. The sandpile identity element on an ellipse: For certain elliptical subsets of the square lattice, the recurrent representative of the identity element of the sandpile group consists predominantly of a biperiodic pattern, along with some noise. It is shown that as the lattice spacing tends to 0, the fraction of the area taken up by the pattern in the identity element tends to 1. Stochastic sandpile on a cycle: In the stochastic sandpile model on a graph, particles interact pairwise as follows: if two particles occupy the same vertex, they must each take an independent random walk step with some probability 0 < p < 1 of not moving. These interactions continue until each site has no more than one particle on it. In this chapter, a formal coupling between the stochastic sandpile and activated random walk models is developed. This coupling is used to show that for the stochastic sandpile with n particles on the cycle graph Z mod n, the system stabilizes in n-cubed time for all initial particle configurations, provided that p(n) tends to 1 suffciently rapidly as n tends to infinity. An urn model for opinion propagation on networks: A coupled Polya's urn scheme for social dynamics on networks is considered. Agents hold continuum-valued opinions on a two-state issue and randomly converse with their neighbors on a graph, agreeing on one of the two states. The probability of agreeing on a given state is a simple function of both of agents' opinions, with higher importance given to agents who have participated in more conversations. Opinions are then updated based on the results of the conversation. I show that this system is governed by a discrete stochastic heat equation, and prove that a consensus of opinion is reached.