I want to discuss some work on large-N limit behavior of random matrices, viewing this as a kind of parallel to Mean Field Games. The description of the limiting object is given by Voiculescu’s free probability theory and the associated stochastic calculus. The limiting object more complicated because of non-commutativity of matrix multiplication. Under some “log-concavity” assumptions it is possible to find analogs of monotone transport theory for the limiting object; in analogy with MFG, one find that the limit object in a certain sense well-approximates the finite-N optimal transport maps. I will describe some joint work with Guionnet and with Guionnet-Dabrowski, as well as work of Figalli and Guionnet, related to these topics. This talk will provide an introduction to David Jekel’s talk later in the day.
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