By Vincent Rivasseau (Chief Editor)

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14) Rn (box) is the euclidean length of J(k), where J : ΛL → ΛL given by J −1 = π|Λ(box) . is the natural bijection L For the derivation of the asymptotics of ETL,β (xj ; xk ), for large d(j − k), the following summability hypothesis, which depends on a weight function G : ΛL → [0, +∞), is an important requirement. Hypothesis 3. [G] For an even function G : ΛL → [0, +∞), there exist weights aij (k) = aji (k) ≥ 0, bij (k) = bji (k) ≥ 0, where i, j, k ∈ ΛL , such that, for all i, j ∈ ΛL and x ∈ R|ΛL | , Hi,j (x) − Hi,j (0) ≤ aij (k)|Hk (x)| + bij (k)|Hk (x)|2 .

1 0 1 ¯ ¯ fq ,fq ∈T (P ) fq ,fq ∈T (βM (a,C) . . β1 )− 4 1 a∈Lu (P )∪Lr0 (P ) − 54 (βM (a,C) . . β1 ) a∈L0 (P ) (βt(i) . . β1 ) (βt(i) . . β1 ) 0c gi ∈DC a∈L1 3 (βM (a,C) . . β1 )− 4 ) 1 2 1c gi ∈DC Each βi appears with the exponent −1 + xi . 99) where we defined ITi0 ITi1 ILi (C) IL0i (C) IL1i (C) Si0 (C) Si1 (C) := := := := := := := {fj , f¯j ∈ T 0 (P )|j ≥ i} {fj , f¯j ∈ T 1 |j ≥ i} {a ∈ L|M (a, C) ≥ i} {a ∈ L0 (P )|M (a, C) ≥ i} {a ∈ L1 |M (a, C) ≥ i} {gj ∈ DC0c (P )|t(j) ≥ i} {gj ∈ DC1c |t(j) ≥ i}.

1 )− 4 1 a∈Lu (P )∪Lr0 (P ) − 54 (βM (a,C) . . β1 ) a∈L0 (P ) (βt(i) . . β1 ) (βt(i) . . β1 ) 0c gi ∈DC a∈L1 3 (βM (a,C) . . β1 )− 4 ) 1 2 1c gi ∈DC Each βi appears with the exponent −1 + xi . 99) where we defined ITi0 ITi1 ILi (C) IL0i (C) IL1i (C) Si0 (C) Si1 (C) := := := := := := := {fj , f¯j ∈ T 0 (P )|j ≥ i} {fj , f¯j ∈ T 1 |j ≥ i} {a ∈ L|M (a, C) ≥ i} {a ∈ L0 (P )|M (a, C) ≥ i} {a ∈ L1 |M (a, C) ≥ i} {gj ∈ DC0c (P )|t(j) ≥ i} {gj ∈ DC1c |t(j) ≥ i}. 100) Vol. 1, 2000 Continuous Constructive Fermionic Renormalization 49 c(i) is the number of connected components in Ti (P ).

### Annales Henri Poincaré - Volume 1 by Vincent Rivasseau (Chief Editor)

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