October 15 
David Jekel
UCLA

Triangular Transport of Measure for Noncommutative Random
Variables
We study tuples $(X_1,\dots,X_d)$ of selfadjoint operators in a
tracial $W^*$algebra whose noncommutative distribution is the free Gibbs
law for a (sufficiently regular) convex potential $V$. Such tuples model
the large $N$ behavior of random matrices $(X_1^{(N)}, \dots, X_d^{(N)})$
chosen according to the measure $e^{N^2 V(x)}\,dx$ on
$M_N(\mathbb{C})_{sa}^d$. Previous work showed that
$W^*(X_1,\dots,X_d)$ is isomorphic to the free group factor
$L(\mathbb{F}_d)$. In a recent preprint, we showed that an isomorphism
$\phi: W^*(X_1,\dots,X_d)$ can be chosen so that $W^*(X_1,\dots,X_k)$ is
mapped to the canonical copy of $L(\mathbb{F}_k)$ inside $L(\mathbb{F}_d)$
for each $k$. The idea behind the proof is to apply PDE methods for
constructing transport to Gaussian to the conditional density of
$X_j^{(N)}$ given $X_1^{(N)}, \dots, X_{j1}^{(N)}$. Then we analyze the
asymptotic behavior of these transport maps as $N \to \infty$ using a new
type of functional calculus, which applies certain
$\\cdot\_2$continuous functions to tuples of selfadjoint operators
to selfadjoint tuples in (Connesembeddable) tracial $W^*$algebras.
