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Example text

K; k= 1,2,... where the sum runs over all non-trivial partitions o f the pair (s, s') into pairs of sets (s^, s p , i = 1,. . ,k , к = I, 2,. . and the func­ tion b(x, x ') on X(Z^ xX (Z ^) is translation invariant b(s, s') = b(s + t, s '+ t), s, s'€:X (Z^), t e z ’^ (2. 5) and satisfies the cluster estimate |b(s, s')| < A C ^^ ^exp {-a L (s U s ') } (2. 6) where A, C, or > O are constant and L(s U s') is the minimal length of a tree joining the points of s U s '. The quantities b(s, s'), s, s' ElX (Z^) are called clusters of the cluster operator A.

22) and also of the subsequent in­ tegrals in (3. 19). REFERENCES J. L Lebowitz, 0 . Penrose, Analytic and clustering properties of thermodynamic functions and distributions functions for classical lattice and continuous systems, Comm. Math. , 11, 99 (1968) and Decay of correlations, Comm. Math. Phys. , 39, 165, (1974). 2. J. L Lebowitz, Bounds of the correlations and analyticity pro­ perties of ferromagnetic Ising spin systems. Comm. Math. , 28, 313-321 (1972). 34 3. F. H, Abdulla-Zadeh, R L Ro A.

4) Si US. , k; k= 1,2,... where the sum runs over all non-trivial partitions o f the pair (s, s') into pairs of sets (s^, s p , i = 1,. . ,k , к = I, 2,. . and the func­ tion b(x, x ') on X(Z^ xX (Z ^) is translation invariant b(s, s') = b(s + t, s '+ t), s, s'€:X (Z^), t e z ’^ (2. 5) and satisfies the cluster estimate |b(s, s')| < A C ^^ ^exp {-a L (s U s ') } (2. 6) where A, C, or > O are constant and L(s U s') is the minimal length of a tree joining the points of s U s '. The quantities b(s, s'), s, s' ElX (Z^) are called clusters of the cluster operator A.