From the GNZ identity to a Dyson–Schwinger cumulant hierarchy for point processes

The Georgii–Nguyen–Zessin identity induces a functional differential equation for the cumulant generating functional of a point process admitting a Papangelou conditional intensity. In generating-functional form, this equation takes the structure of a Dyson–Schwinger identity. For pairwise Gibbs point processes, the equation closes: insertion of a point acts as a deterministic shift of the source field. This yields a hierarchy for cumulant densities of all orders, expressing cumulants as differences between the original law and the laws induced by point insertion. The formulation is non-perturbative and provides exact recursion relations for cumulants.