Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation

We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for $\phi^4$ theory in 2D. It is based on an exact treatment of the functional derivative $\partial \Gamma / \partial G$ of the 4-point vertex function $\Gamma$ with respect to the 2-point correlation function $G$ within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around $G=0$ in a HAM control parameter $c_0G$, or even a convergent one up to the phase transition point if shifts in $G$ can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving integro-differential equations.