## Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results.(English)Zbl 1292.47016

Let $$\varphi$$ be an integrable function on the unit circle with the Fourier coefficients $$\varphi_k$$. The Toeplitz matrices $$T_n(\varphi)$$ and Toeplitz determinants $$D_n(\varphi)$$ with symbol $$\varphi$$ are given by $$T_n(\varphi)=\{\varphi_{j-k}\}_{0\leq j,k\leq n-1}$$ and $$D_n(\varphi)=\det T_n(\varphi)$$, respectively. The authors say that “our goal in this paper is not to review the development and applications in a comprehensive way, but rather to focus on one central problem in the theory of Toeplitz determinants, the Szegő strong limit theorem”. This theorem says that, if $$\varphi$$ is a positive function in $$C^{1+\varepsilon}$$ with $$\varepsilon>0$$, then $\lim\limits_{n\to\infty}{D_n(\varphi)\over e^{n(\log\varphi)_0}}=e^{E(\varphi)},$ where $$E(\varphi)=\sum_{k=1}^\infty k|(\log\varphi)_k|^2$$.
In the survey paper under review, the history of the Szegő strong theorem and its further generalizations is discussed in detail in connection with questions raised in the analysis of the two-dimensional Ising model, one of the central models in statistical mechanics. This model concerns the interaction of random spins $$\sigma_{i,j} =\pm 1$$ at sites $$(i,j)$$ in the lattice $${\mathbb Z}^2$$. The situation of the so-called ferromagnetic system, i.e., when parallel spins have lower energy than antiparallel spins, is of great interest. This happens, in particular, when only nearest-neighbor spins interact and the interaction energy is given by $$-J_1\sigma_{i,j}\sigma_{i,j+1}-J_2\sigma_{i,j}\sigma_{i+1,j}$$, where the vertical and horizontal interaction constants, $$J_1$$ and $$J_2$$ respectively, are translation invariant and positive.
The paper is nicely written and is addressed to a wide audience. Specialists in many areas will enjoy an abundance of factual material and many interesting historical details. The list of references is up to date, it contains 199 items.

### MSC:

 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 15B05 Toeplitz, Cauchy, and related matrices 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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