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Denote by JlA',xA' the projection on X(A'), A' c A of the probability Jl xA" Denote by PZ',A the corresponding probability density. Clearly, the following formula holds: P~~,(XA') = (2ntklA'I/2(det Cf)1/2exp(-~ I (C~{(Xt - Xt ), (x. - Xs))). 2. 1. 5) (also on v, k, r, but we fix these values throughout the whole section) such that the following estimates hold: (a) For any finite A c 71", any la~,ul SE A and UE « K qls-u 1. 15) (b) For any finite A c 71", any s, tE A, Ib~,tl « Kqls-t l. 17) 20 R.

M} such that there is a oneto-one correspondence between bonds {i,j} E T and bonds {s, t} E r, such that s E Si and t E Sj. 40) (the number of bonds of r contamlng the vertex t). Then all the moments

L. DOBRUSHIN AND M. 32) b~,t A ;;;. cE, where E is the identity matrix, in the sense of inequalities between positive definite matrices. 32) follows. 16) (with a different K). 17). 33) where L, M are some k IA' I x k IA" A'I matrices. 34) Therefore, L = -B~' M(A'A')-1 = -tB~' M BA,A' (because BA,A' (B~') -1 = t(A'A')-l) and = 2<1>A' - MBA'A'MT. 36) for each s, tEN. Now we use the fact that has a finite range r. Let K' > 0 be choosen such that 1<1>/1 < K', t E lLv.