We study efficient and stable mechanisms in matching markets when the number of agents is large and individuals’ preferences and priorities are drawn randomly. When agents’ preferences are uncorrelated, then both efficiency and stability can be achieved in an asymptotic sense via standard mechanisms such as deferred acceptance and top trading cycles. When agents’ preferences are correlated over objects, however, these mechanisms are either inefficient or unstable even in an asymptotic sense. We propose a variant of deferred acceptance that is asymptotically efficient, asymptotically stable and asymptotically incentive compatible. This new mechanism performs well in a counter- factual calibration based on New York City school choice data.