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**Extra info for Communications in Mathematical Physics - Volume 198**

**Example text**

42). We have |E1(k+1) (z, Z, Y0 )| ≤ D⊂D Y ∈D 3 1 ε 2 exp(−(2κ − 1 − 5κ0 )dk+1 (Y )). 44) Y ∈D · dψ Z χ(k) (Z) exp − 1 ψ 2 2 Z (k) (Z, τ )|−1 . sup |Zs,0 D,τ Consider the product over Y ∈ D of the exponential factors above, and take a graph whose vertices are domains of the set D ∪ {Y0 }, and whose edges are pairs of the domains {Y , Y } such that their intersections are non-empty, Y ∩ Y = ∅. It is a connected graph, because the domain Z is connected. Take a connected subgraph of this graph, which is a tree graph, and which has the same set of vertices D ∪ {Y0 }.

Phys. 182, 675–721 (1997) Renormalization and Localization Expansions 45 5. : A Low Temperature Expansion for Classical N -Vector Models. III. A Complete Inductive Description, Fluctuation Integrals. Commun. Math. Phys. 182, 675–721 (1997) 6. : Commun. Math. Phys. a) 89, 571–597 (1983); b) 119, 243–285 (1988); c) 122, 175–202 (1989); d) 122, 355–392 (1989) 7. : A Short course in Cluster Expansions. In Critical Phenomena, Random Systems, Gauge Theories. Les Houches (1984), London–New York: Elsevier Science Publishers, 1986 8.

See one of the references [7,9] for a defi1 nition of the functions ρT . 46)| > e−1 exp(−dk+1 (Z)). 51) This completes the estimate of the function E1(k+1) (z, Z, Y0 ), and we obtain |E1(k+1) (z, Z, Y0 )| < e2 ε exp(−6κ0 dk+1 (Y0 )) exp(−(2κ − 3 − 6κ0 )dk+1 (Z)). 53) E1(k+1) (z, Z, Y0 ). 1. 25) properly restricted, and with Y replaced by Z, and they satisfy the bounds |E1(k+1) (z, Z)| < e2 K0 ε exp(−(2κ − 3 − 6κ0 )dk+1 (Z)). 54) Let us notice that the constant e2 K0 ε is already quite small by all the assumptions on ε we have introduced.