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$$\langle\sigma_{0,0}\sigma_{M,N}\rangle_{\pm}\ .$$ For example we could take Zd, the set of points in Rd all of whose coordinates are integers. The case T>Tc. This is an extremely deep 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. We find that the leading approach to $${\mathcal M}_{-}^2$$ These exact calculations have given microscopic insight into the many body collective phenomena of phase transitions and have developed new areas of mathematics. To begin with we need a lattice. An extremely powerful method which has been generalized to solve many classes of models in 2 dimensions was invented by R.J. Baxter which relies on an invariance exhibited by the anisotropic model where The study of the spin-spin correlations for large separations was initiated in 1966 by T.T. In 1949 Kaufman found a much simpler method of computing the free energy and the partition function by use of spinor analysis. The phenomenology of scaling theory originates in these computations. where $$\sigma_{j,k}$$ is the spin in row $$j$$ and column $$k\ ,$$ The Hamiltonian, H of the Ising model is give by: *= − , Í O Ü O Ý Ü. Ý J ij is the interaction energy between spins at lattice point i and j. In this scaling limit we define scaling functions for $$T$$ above and below $$T_c$$ as, $$G_{\pm}(r)=\lim_{ {\rm scaling} }{\mathcal M}_{\pm}^{-2}\langle\sigma_{0,0}\sigma_{N,N}\rangle_{\pm}$$. The leading behavior as $$N\rightarrow \infty$$ is given by the large $$N$$ behavior of $$f^{(1)}_N(t)$$, $$\langle\sigma_{0,0}\sigma_{N,N}\rangle_{+}\sim(1-t)^{1/4}f^{(1)}_N(t)=\frac{t^{N/2}}{(\pi N)^{1/2}(1-t)^{1/4}}+\cdots$$. McCoy, C.A. The Ising model is unique among all problems in statistical because not only can $$\sinh 2E^v/k_BT\sinh2E^h/k_BT$$ The terms $$t^{N}$$ and $$t^{N/2}$$ are often written as, $$t^{N/2}=e^{(N/2)\ln t}$$ and $$t^N=e^{N\ln t}$$, where $$\xi$$ is called the correlation length. defined by, $$\chi_d(T)=\frac{1}{k_BT}\sum_{N=-\infty}^{\infty}\{\langle \sigma_{0,0}\sigma_{N,N}\rangle \frac{ {\mathcal M}^2_{+} }{k_BT}\sum_{n=0}^{\infty}{\hat\chi}^{(2n+1)}(T)$$, where the explicit expressions for $${\hat\chi}^{(j)}(T)$$ for general mean zero Gaussian variables with variance $\epsilon^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. Onsager's 1944 computation computes the free energy by discovering that the model can be described by what is now known as an $$sl_2$$ loop algebra. the determinants of Kaufman and Onsager for the two spin correlation function when the separation Prof. Barry McCoy, Stony Brook University, USA. The amplitudes $$A_{\pm}$$ are different for $$T$$ above or below $$T_c$$ and the ratio $$A_{+}/A_{-}$$ is approximately $$12\pi\ .$$ Orrick, B.G Nickel, A.J. mathematical development, the full generality of which is still under development. Orrick and N. Zenine (2007). \int_0^1\prod_{k=1}^{2n+1}x_k^Ndx_k Exact results may also be obtained if the coupling $$E^v$$ with, $$G_{\pm}(r)=\lim_{ {\rm scaling} }{\mathcal M}_{\pm}^{-2}\langle \sigma_{0,0}\sigma_{M,N}\rangle_{\pm}$$. $$T>T_c\ .$$, 1, The case T=Tc. Wu in 1967 for the properties of spins near the boundary of a half plane with a magnetic field $$H_b$$ interacting with the row of boundary spins $$\sigma_{1,m}\ .$$ The spontaneous magnetization of a spin on the boundary is, $${\mathcal M}_b=\left[\frac{\cosh 2K^v-\coth2K^h}{\cosh 2K^v-1}\right]^{1/2}$$, Thus the critical exponent $$\beta=1/2$$ for the boundary magnetization. $$T\rightarrow T_c$$) this correlation length diverges as, The large $$N$$ behaviors of the two point function for fixed As $$t\rightarrow 1$$ (2\pi m/j)=0\), For $$T>T_c$$ the singularity in $${\hat\chi}^{(2n+1)}(T)$$ is of the form, $$A_{2n+1}\epsilon^{2n(n+1)-1}\ln \epsilon$$, and for \(T