Spectra of stochastic operators arising in lattice models of a gas

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.