## 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

### Keywords:

Curie-Weiss-Ising model; free energy