By Vincent Rivasseau (Chief Editor)

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N ! 96) 48 M. Disertori and V. Rivasseau Ann. Henri Poincar´ e where by convention u0 = 1. The Jacobian of this transformation is the determinant of a triangular matrix hence it is given by: n ¯ −1 J = β1 (β1 β2 ) . . (β1 β2 . . βn¯ −2 ) = βin¯ −1−i . 97) i=1 We absorb Λ−n into the term K n¯ since we recall that Λ > m hence that Λ−n = (Λm )−n , and that in Theorem 3 Λm remains in the compact X, hence is bounded away from 0. 95) becomes 2p ||φi ||∞,2 e−(1− |ΓΛ 2p (φ1 , . . n ! −1+ 12 (¯ n−i)− 12 |Ni | βi i=1 ¯ −1 1n 1 ...

Vol. 1, 2000 Continuous Constructive Fermionic Renormalization 51 n−2 ¯ Moreover, we bound (¯n1)! u−T f (T ) by n¯ n¯ ! supu−T |f (T )| using Cayley’s theorem. Therefore, by Stirling’s formula, it’s enough to consider the unordered tree T which gives the maxu−T |f (T )|. The sum that could still give some factorial is σ . To bound it we use the product of fractions obtained after integration on the βi . 113) Now |ETi | depends on the (now unordered) tree T and on its ordering σ. Therefore we call it from now on |ETiσ |.

40), for some r > 1, we assume that Hypothesis 3[G] holds with G ≡ θrS , where ˜ θrS (k) := inf Srv (k) k˜ ∈ π −1 (k) . 18) by dS (x) := inf x˜∈π−1 (x) {S(˜ x)} defines a semi-metric which obeys the triangle inequality, dS (x+y) ≤ dS (x)+dS (y). The choice of G in the three cases decribed above can be expressed in terms of the underlying semi-norm on Rn , namely, G ≡ dS ≡ 0, for S ≡ 0, G ≡ dS = νd, for S = ν| · |Rn , and G ≡ dS = θrS , for S ≡ Srv . We remark that Hypotheses 1 and 3 partially strengthens Hypothesis 2, as they imply that there is only one critical point, namely at the minimum.