The network interference model for treatment effect estimation places experimental units at the vertices of an undirected exposure graph, such that treatment assigned to one unit may affect the outcome of another unit if and only if these two units are connected by an edge. This model has recently gained popularity as means of incorporating interference effects into the Neyman-Rubin potential outcomes framework; and several authors have considered estimation of various causal targets, including the direct and indirect effects of treatment. In this paper, we consider large-sample asymptotics for treatment effect estimation under network interference in a setting where the exposure graph is a random draw from a graphon. When targeting the direct effect, we establish a central limit theorem and find that-in our setting-popular estimators are considerably more accurate than existing results suggest. Meanwhile, when targeting the indirect effect, we leverage our generative assumptions to propose a consistent estimator in a setting where no other consistent estimators are currently available. Overall, our results highlight the promise of random graph asymptotics in understanding the practicality and limits of causal inference under network interference.