Probabilistic Operator Algebra Seminar: Triangular Transport for Free Gibbs Laws from Convex Potentials

Seminar | September 16 | 3-5 p.m. | 736 Evans Hall

 David Andrew Jekel, UCLA

 Department of Mathematics

We study tuples $(X_1,\dots ,X_m)$ of self-adjoint operators in a tracial $W^*$-algebra whose non-commutative distribution is free the Gibbs law for a (sufficiently regular) convex potential $V$. Such tuples model the large $N$ behavior of random matrices $(X_1^{(N)}.\dots ,X_m^{(N)}$ chosen according to the measure $e^{-N^2 V(x)}\,dx$ on $M_N( \mathbb C)_{sa}^m$. Previous work showed that $W^*(X_1,\dots ,X_m)$ is isomorphic to the free group factor $L(\mathbb F_m)$. In a recent preprint, we showed that an isomorphism $\phi :W^*(X_1,\dots ,X_m)$ can be chosen so that $W^*(X_1,\dots ,X_k)$ is mapped to the canonical copy of $(\mathbb F_k)$ inside $L(\mathbb F_m)$ for each $k$. The idea behind the proof is to apply PDE methods for constructing transport to Gaussian to the conditional density of z$X_j^{(N)}$ given $X_1^{(N)},\dots , X_{j-1}^{(N)}$. Then we analyze the asymptotic behavior of these transport maps as $N \to \infty $ using a new type of functional calculus, which applies certain $ norm (\cdot )_ 2$-continuous functions to tuples of self-adjoint operators to self-adjoint tuples in (Connes-embeddable) tracial $W^*$-algebras.