This paper provides a model of social learning where the order in which actions are taken is determined by an m-dimensional integer lattice rather than along a line as in the herding model. The observation structure is determined by a random network. Every agent links to each of his preceding lattice neighbors independently with probability p, and observes the actions of all agents that are reachable via a directed path in the realized social network. For m ≥ 2, we show that as p < 1 goes to one, (i) so does the asymptotic proportion of agents who take the optimal action, (ii) this holds for any informative signal distribution, and (iii) bounded signal distributions might achieve higher expected welfare than unbounded signal distributions. By contrast, if signals are bounded and p=1, all agents select the suboptimal action with positive probability.