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Estimates of semi-invariants for the Ising model at low temperatures. (English) Zbl 0873.60074
Dobrushin, R. L. (ed.) et al., Topics in statistical and theoretical physics. F. A. Berezin memorial volume. Transl. ed. by A. B. Sossinsky. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 177(32), 59-81 (1996).
The cluster expansion method is one of the most powerful mathematical tools in the study of mathematical physics. See for instance the book “Gibbs random fields. Cluster expansions.” Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0731.60099) by V. A. Malyshev and R. A. Minlos. The author presents a new variant of the method. As a consequence, the author proves the following result: For lower enough temparature, the \(r\)-th order semi-invariants in a finite volume are decayed by an exponential rate, which is the product of a constant depending on the inverse temperature only and the \((d-1)\)-dimensional area of the minimal contour containing the whole finite set. Moreover, the upper bound is a geometric characteristic quantity to the power \(|r|\). The key point is that the known estimate for higher temperature is not available for the present situation. The approach here was indeed used before in some special cases by the author and his coauthors in previous publications.
For the entire collection see [Zbl 0853.00022].

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics