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

**Sample text**

We will discuss it again later, when we will have all conditions on β and |σ|. 45)[3] over the domains Y1 determining a fixed domain Y ∈ Dk+1 ( k+1 ). 18) ):X1 ⊂Y k+1 The terms of this expansion are analytic functions of (θ, h) on the spaces c (Y, Bk+1 ; {εj }). 15). 129)[1], or to E0 for j > k − n1 . Let us recall also that X1 is a domain in Dk+1 containing the domain X. It is defined as a union of L2 M -cubes from the cover πk+1 , containing an LM Lj η-cube from the cover πj contained in X. 18), and then we do the partial resummation of the terms with a fixed localization domain Y ∈ Dk+1 ( k+1 ).

22) and the sum is over x ∈ j = 1, . . , k, X ∈ Dj such that x ∈ Xand X ⊂ (Zkc )≈ . 20) replaced by a sum over x ∈ Y ∩ T1 . 20), and of g defined on the space {g : g is defined on T1 ∩ Y and has values in Cn , g 1 < 1}. 21). 1). 25) depending on these variables restricted to the domain Y . 23). The positive number α1 above can be chosen as arbitrarily small. 17), but now we assume that the expression in front of the exponential 32 T. Balaban above is sufficiently small. β − 4 +α1 ≤ 1, β − 8 η γ−α1 |σ| ≤ 1.

Math. Phys. 175, 607–642 (1996) 3. : Localization Expansions. I. Functions of the “Background” Configurations. Commun. Math. Physics 182, 33–82 (1996) 4. : A Low Temperature Expansion for Classical N -Vector Models. II. Renormalization Group Equations. Commun. Math. 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.