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1)) We will construct, in the forthcoming Definitions 1 and 2, a perturbation of such Gaussian fields, of the following type. DEFINITION 1. Lv (the support of A) is assigned. Denote by N(A) the number of A' E 21 such that supp A' = supp A. NOTE. Lv, with supp A = A. 4. DEFINITION 2. Suppose that some symmetric relation )( (compatibility) is given on 21. Suppose that the relation )( is such that for some integer d, dist(supp A, supp A') ;;. 101) A)(A';A,A' E 21. We say that a subsystem d c 21 is compatible if A)( A' for each A, A' E d.

24) (cf. ) imply that = 0 whenever ~ji < ~j; (cf. 7)) and also = whenever ~ji = ~j;. 18). 17). 25) o The following lemma (which will be frequently referred to later) is an example of the diagrammatic method in expressions of semi-invariants of (products of) Wick polynomials. The method is standard, but we discuss it in some detail here because of lack of suitable references. LEMMA 3. ,n(i,j). 26) 1 Denote by f (F) the set of all vacuum diagrams g; on the set F of all triples (i,j,k).

36). '/\' ",V I . 3. The Case of Infinite A Let A c 7L v. 43) 00 for any monotone sequence Al c Az C ... such that u;;'; 1 An = 7L v, An is finite for each n. This limit is clearly independent of the choice of {An}. 24 R. L. DOBRUSHIN AND M. ZAHRADNIK PROPOSITION. 2 remain true even for infinite A. 2. NOTE 1. In the special case I\. ,t>c\~( instead of bft> cf('. ,t depend on Is - tl only. - t . 2). 44) s,tEA where I\. }-l. 45) COROLLARY. 5) such that (a) Ib,l < Kqlsl for any 7L '. 46) any S,tEI\..

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