Toy Models of Non-perturbative Asymptotic Freedom in $\phi^3_6$

We study idealizations of the full nonlinear Schwinger-Dyson equations for the asymptotically free theory of $\phi^3$ in six dimensions in its meta-stable vacuum. We begin with the cubic nonlinearity and go on to all-order nonlinearities which contain instanton effects. We show how our toy models of the cubic Schwinger-Dyson equations contain the usual diseases of perturbation theory in the massless limit (e.g., factorially divergent $\beta-$functions, singular Borel-transform kernels associated with infrared renormalons) and show how these models yield specific mechanisms for removing such singularities when there is a mass gap. In the all-order nonlinear equation we show how to recover the usual renormalization-group-improved instanton effects and associated factorial divergences.