Geodesic
Thermodynamical averages for the~Ising model and spectral invariants of Toeplitz matrices
Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 3, pp. 401-416

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We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of $N$ particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large $N$, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable $\beta$ $(\beta$ is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as $N\to\infty$.
Keywords: Ising model, statistical model, specific free energy, asymptotics, Toeplitz matrix. 
@article{TMF_2020_203_3_a5,
     author = {V. M. Kaplitskii},
     title = {Thermodynamical averages for {the~Ising} model and spectral invariants of {Toeplitz} matrices},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {401--416},
     publisher = {mathdoc},
     volume = {203},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2020_203_3_a5/}
}
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V. M. Kaplitskii. Thermodynamical averages for the~Ising model and spectral invariants of Toeplitz matrices. Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 3, pp. 401-416. http://geodesic.mathdoc.fr/item/TMF_2020_203_3_a5/