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##### Department of Mathematics, University of California San Diego

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## Transport equations in random matrices and non-commutative probability

##### Abstract:

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.

### APM 6402

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