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Thermodynamics of the anisotropic Curie-Weiss-Ising model. (Russian) Zbl 0578.60098

Teor. Veroyatn. Mat. Stat. 31, 13-19 (1984).
Let \(T_{MN}\) be the finite subset \(\{\) (i,j): \(1\leq i\leq M\), \(1\leq j\leq N\}\) of \({\mathbb{Z}}^ 2\). Given a Hamiltonian \[ H(\sigma_{MN})=- \sum_{(i,j),(i',j')\in T_{MN}}\rho_{MN}(i,j;i',j')s_{ij}\sigma_{i'j'}-h\sum_{(i,j)\in T_{MN}}\sigma_{ij} \] where \(\sigma_{ij}\) have values in \(\{\pm 1\}\), \(h\) is a parameter and \[ \rho_{MN}(i,j;i',j')= \begin{cases} I_ 1/(2M) &\quad\text{ if }j=j'; \\ I_ 2/2 &\quad\text{ if }i=i',\;| j-j'| =1;\text{ and }\\ 0 &\quad \text{otherwise,}\end{cases} \] with positive \(I_ 1\) and \(I_ 2.\) Define \[ Z_{MN}= \sum_{\{\sigma_{MN}\}} \exp\{-\beta H(\sigma_{MN})\},\quad \beta >0. \] The author shows that the free energy has the form: \[ \psi(\beta,h)\equiv \beta^{-1}\lim_{M,N\to \infty}(MN)^{-1}l_ nZ_{MN}=-\beta^{-1}[l_ n(2ch(\beta I_ 1))+ \]
\[ \max_{| x| <1}\phi(x)]=-\beta^{-1}[l_ n(2ch(\beta I_ 1))+\phi(x_ 0) \] where \(\phi(x)=2^{-1}\ell n(\frac{1-r^ 2x^ 2}{1-x^ 2})-(16\beta I_ 2)^{-1}[\ell n(\frac{1+x}{1-x}\cdot \frac{1-rx}{1+rx})-2\beta h]^ 2,\)
\(r=th(\beta I_ 2)\), \(x_ 0=x_ 0(\beta,h)\) is the root of \(4\beta I_ 1(1+r)/(1+rx^ 2)=\ell n((1+x)(1-rx)/(1-x)(1+rx))\) which achieves the maximum of h. In particular, if \(h=0\) then \[ \psi (\beta,0)=\begin{cases} -\ell n(2ch(\beta I_ 2))/\beta,&\quad\text{ if }\beta \leq \beta_ c\\ -[\ell n(2ch(\beta I_ 2)+ \phi(x_ 0)]/\beta,&\quad\text{ if }\beta >\beta_ c,\end{cases} \] where \(\beta_ c\) is determined by \(2\beta I_ 1 \exp(2\beta I_ 2)=1\), and \(x_ 0\) is now positive.
Reviewer: M.Chen

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory