We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.
Spectral density of dense random networks and the breakdown of the Wigner semicircle law