Spectra of stochastic operators arising in lattice models of a gas
Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 2, pp. 230-243
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The authors investigate the spectrum of the transfer-matrix $A_L$ for general lattice models with finite interaction. For this purpose they construct the limiting stochastic operator $P_\infty$, which is the limit of the stochastic matrices $P_L$ obtained by a natural normalization from transfer-matrix $A_L$. For the operator $P_\infty$ for small values of $\beta$ they find the first two invariant subspaces, on one of which the spectrum of operator $P_\infty$ coincides with the values of a certain function $a(\lambda)$ $(0<\lambda<2\pi)$, while in the other it contains values of the function $a(\lambda_1)a(\lambda_2)$ $(0<\lambda_1<\lambda_2\leqslant 2\pi)$, and also, perhaps, a number of segments. This result agrees with the well-known work of Onsager in which the spectrum of $P_\infty$ was calculated in explicit form for a particular case of the Ising model.
@article{TMF_1970_2_2_a5,
author = {R. A. Minlos and Ya. G. Sinai},
title = {Spectra of stochastic operators arising in lattice models of a~gas},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {230--243},
year = {1970},
volume = {2},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1970_2_2_a5/}
}
R. A. Minlos; Ya. G. Sinai. Spectra of stochastic operators arising in lattice models of a gas. Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 2, pp. 230-243. http://geodesic.mathdoc.fr/item/TMF_1970_2_2_a5/
[1] N. N. Bogolyubov, B. I. Khatset, DAN SSSR, 66 (1949), 321 | MR | Zbl
[2] R. L. Dobrushin, Teoriya veroyatn. i ee prim., 13 (1968), 201
[3] R. L. Dobrushin, Funkts. analiz i ego prilozh., 2 (1968), 31 | Zbl
[4] M. Kac, C. Thompson, Proc. Nat. Acad. Sci., USA, 55 (1966), 676 | DOI | MR | Zbl
[5] B. Kaufman, Phys. Rev., 76 (1949), 1232 | DOI | Zbl
[6] R. A. Minlos, Funkts. analiz i ego prilozh., 1 (1967), 60 | MR | Zbl
[7] L. Onsager, Phys. Rev., 65 (1944), 117 | DOI | MR | Zbl
[8] D. Ruelle, Preprint IHES, 1966 | MR