A sorting network is a shortest path from the identity to the reverse permutation in the Cayley graph of S_n generated by adjacent transpositions. Remarkable conjectures about the global scaling limit of a uniformly random sorting network have been made based on strong empirical evidence.

One approach to proving these conjectures is to first show the existence of a local limit of random sorting networks, and then use this to piece together global information. In this talk, I will discuss this local limit and progress that has been made towards understanding the global limit as a consequence of local properties.