Free transport for interpolated free group factors

In this article, we study a form of free transport for the interpolated free group factors, extending the work of Guionnet and Shlyakhtenko for the usual free group factors. Our model for the interpolated free group factors comes from a canonical finite von Neumann algebra $\mathcal{M}(\Gamma, \mu)$ associated to a finite, connected, weighted graph $(\Gamma,V,E, \mu)$. With this model, we use an operator-valued version of Voiculescu's free difference quotient to state a Schwinger-Dyson equation which is valid for the generators of $\mathcal{M}(\Gamma, \mu)$. We construct free transport for appropriate perturbations of this equation. Also, $\mathcal{M}(\Gamma, \mu)$ can be constructed using the machinery of Shlyakhtenko's operator-valued semicircular systems