I will present recent progress on developing a Wasserstein information geometry for free probability, which "should" describe the large-$n$ behavior of Wasserstein information geometry for random (multi) matrix models. In particular, I will focus on whether the analog of displacement concavity of entropy holds in the free probability setting, and the related question of whether the free entropy and free Wasserstein distance actually give the large-$n$ limit of the quantities for matrix models. I will show that for certain $X$ and $Y$, the same random matrix models cannot asymptotically realize the desired entropy and Wasserstein distance simultaneously, due to their not accounting for additional information about how $X$ interacts with the ambient algebra. Moreover, I use continuous model theory to provide a new framework which allows for reasonable estimates of entropy along Wasserstein geodesics.
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