Uniqueness of gibbs fields via cluster expansions

For the unbounded spin systems one cannot get cluster expansion if there exist large enough boundary values. A simple idea to avoid these difficulties is to prove that with probabilityp 1 when ^v there is a large subvolume of such that on all spin values do not exceed some fixed number. This gives a new method to prove uniqueness results for the unbounded spin systems generalizing some results of Refs. 1 and 2. The formulations of these results are in Section 1; the proofs are in Section 2.