Explicit fermionic tree expansions

We express connected Fermionic Green's functions in terms of two totally explicit tree formulas. The simplest and most symmetric formula, the Brydges-Kennedy formula is compatible with Gram's inequality. The second one, the rooted formula of Abdesselam and Rivasseau, respects even better the antisymmetric structure of determinants, and allows the direct comparison of rows and columns which correspond to the mathematical implementation in Grassmann integrals of the Pauli principle. To illustrate the power of these formulas, we give a 'three lines proof' that 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.