EXPLICIT FERMIONIC TREE EXPANSIONS

We express connected Fermionic Green's functions in terms of two total ly explicit tree formulas. The simplest and most symmetric formula, th e Brydges-Kennedy formula is compatible with Gram's inequality. The se cond one, the rooted formula of Abdesselam and Rivasseau, respects eve n better the antisymmetric structure of determinants, and allows the d irect comparison of rows and columns which correspond to the mathemati cal implementation in Grassmann integrals of the Pauli principle. To i llustrate the power of these formulas, we give a 'three lines proof' t hat the radius of convergence of the Gross-Neveu theory with cutoff is independent of the number of colors, using either one or the other of these formulas.