Free monotone transport
By Alice Guionnet, D. Shlyakhtenko. 2012
84 citations recorded by the source corpus.
By solving a free analog of the Monge-Ampère equation, we prove a non-commutative analog of Brenier’s monotone transport theorem: if an n-tuple of self-adjoint non-commutative random variables Z1,…,Zn satisfies a regularity condition (its conjugate variables ξ1,…,ξn should be analytic in Z1,…,Zn and ξj should be close to Zj in a certain analytic norm), then there exist invertible non-commutative functions Fj of an n-tuple of semicircular variables S1,…,Sn, so that Zj=Fj(S1,…,Sn). Moreover, Fj can be chosen to be monotone, in the sense that and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C∗(Z1,…,Zn)≅C∗(S1,…,Sn) and $W^{*}(Z_{1},\dots,Z_{n})\cong L(\mathbb{F}(n))$. Thus our condition is a useful way to recognize when an n-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $\varGamma_{q}(\mathbb{R}^{n})$ are isomorphic (for sufficiently small q, with bound depending on n) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.
Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/ed988ee5960d35d3b03ff68df3fcee4f7d6d1d37
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Citations in the corpus
- Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices (2012, background): “The recent paper of Guionnet and Shlyakhtenko [GS12] gives some basis and highlights many pitfalls of this theory.”
- Free Probability and Random Matrices (2014, background): “So we are, for example, not addressing such exciting topics as free stochastic and Malliavin calculus [40, 108, 114], or the rectangular version of free probability [28], or the strong version of asymptotic freeness [48, 58, 89], or free monotone transport [83], or the relation with representation t”