A related method which reduces the computation of the partition function and correlation functions to a graph counting problem was given by Kac, Ward, Potts, Hurst and Green in several papers, and the relation to a solvable problem in dimer statistics was presented by P.W. Wu (1966), Theory of Toeplitz determinants and the spin correlations of the two dimensional Ising model, Physical Review 149, 380-401. The leading behavior as \(N\rightarrow \infty\) is given by the large \(N\) behavior of \(f^{(1)}_N(t)\), \(\langle\sigma_{0,0}\sigma_{N,N}\rangle_{+}\sim(1-t)^{1/4}f^{(1)}_N(t)=\frac{t^{N/2}}{(\pi N)^{1/2}(1-t)^{1/4}}+\cdots\). The size of these determinants grows with the separation between the spins. =N^2\left((t-1)\frac{d\sigma}{dt}-\sigma\right)^2- \left(\frac{\prod_{j=1}^n x_{2j}(1-x_{2j})(1-tx_{2j})} depends only on \(r\) and not on the ratio \(\sinh 2K^v_c/\sinh2K^h_c\) where \(\lambda=1\) This is the method used in 1976 by Wu, McCoy, Tracy and Barouch. the determinants of Kaufman and Onsager for the two spin correlation function when the separation In 1980 a corresponding characterization of the diagonal correlation function for all temperatures in terms of a Painlevé VI function was found by M. Jimbo and T. Miwa, \(\left(t(t-1)\frac{d^2\sigma}{dt^2}\right)^2 E^v\sigma_{j,k}\sigma_{j+1,k}+H\sigma_{j,k}\}\). THE 2D ISING MODEL 1/2 Figure 2.4: A polygon configuration contributing to the high-temperature expansion of the partition function Z. where the specific heat \(c\) diverges as \(T\rightarrow T_c\) as, and thus the critical exponent \(\alpha=0\ .\) Onsager's algebra. The final thermodynamic property of interest is the magnetic susceptibility at \(H=0\), which is expressed in terms of \(\langle\sigma_{0,0}\sigma_{M,N}\rangle\) as, \(\chi(T)=\frac{1}{k_BT}\sum_{M=-\infty}^{\infty}\sum_{N=-\infty}^{\infty}\{\langle\sigma_{0,0} However if the coupling constants \(E^v(j)\) are chosen randomly out of a probability distribution it was discovered by B.M. To obtain an interpolating function between the regimes we need to consider the case where, \(N\rightarrow \infty\) and \(t\rightarrow 1\ .\). and in 1963 by E.W. where \(K(t^{1/2})\) (and \(E(t^{1/2})\)) is the complete elliptic integral of the first (second) kind. Wu in 1968 that the specific heat is finite at the critical temperature and that the logarithmic singularity of the nonrandom lattice has become an infinitely differentiable essential singularity. Local distributions of the 1D dilute Ising model Yu.D. the full mathematical structure of the susceptibility as a function of temperature is as yet unknown. Wu, B.M McCoy, C.A.Tracy and E. Barouch (1976). and there an interaction energy between nearest neighbor spins of \(-E\) if the We find that the leading approach to \({\mathcal M}_{-}^2\) The Ising model is unique among all problems in statistical because not only can \(\sigma\) which take on the values \(\pm 1\) on the sites of a lattice It is nonzero only for \(T.

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