we investigate the analogs of optimal transport theory in the setting of multivariable asymptotic random matrix theory.  asymptotic random matrix theory concerns the behavior of randomly chosen $n \times n$ matrices in the limit as $n \to \infty$.  for several random matrices $x_1^{(n)}$, $\dots$, $x_d^{(n)}$, one can study the asymptotic behavior of expressions like $(1/n) \tr(x_{i_1} \dots x_{i_k})$, and the appropriate limiting object is a non-commutative probability space, that is, a von neumann algebra $a$ of "random variables" together with an expectation map $e: a \to \mathbb{c}$, analogous to the expected trace of a random matrix.  meanwhile, optimal transport theory asks for the most efficient way to rearrange one distribution of mass $\mu$ on $\mathbb{r}^d$ into another such distribution $\nu$.  such a scheme is often given by transporting the mass at point $x$ to point $f(y)$, for a smooth function $f: \mathbb{r}^d \to \mathbb{r}^d$.

optimal transport is more challenging to make sense of in the non-commutative setting because, unlike classical probability theory, there are many non-isomorphic atomless non-commutative probability spaces, and in fact, space of non-commutative probability distributions fails basic separability and finite-dimensional approximation properties that one is used to in classical probability.  so there is often no possibility of transporting given non-commutative probability distribution $\mu$ to $\nu$ by some map $f$; nonetheless, for the relaxed problem of optimal couplings, we can recover a non-commutative analog of the monge-kantorovich duality characterizing optimal couplings. furthermore, in the regime of convex free gibbs laws (an analog of smooth log-concave probability measures on $\mathbb{r}^d$), non-commutative transport can be achieved by non-commutative smooth functions obtained as solutions to differential equations much like the classical case.  moreover, the non-commutative analog of triangular transformations of measures led to new insight into the structure of the underlying von neumann algebras.