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Register a new userForm factor expansion of the row and diagonal correlation functions of the two dimensional Ising model
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We derive and prove exponential and form factor expansions of the row correlation function and the diagonal correlation function of the two dimensional Ising model.
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In this note we consider non-equilibrium steady states of one-dimensional models of heat conduction (wealth exchange) which are coupled to some reservoirs creating currents. In particular we will give sufficient and necessary conditions which will depend only on the first two moments of the reservoir measures and the redistribution parameter under which the two-point functions are multilinear. This presents the first example of multilinear two-point functions in the absence of product stationary measures.
We prove the analogue of the strong Szeg{H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk cite{YP} for the next-to-diagonal correlations $langle sigma_{0,0}sigma_{N-1,N} rangle$ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.
We study the Ising model two-point diagonal correlation function $ C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $lambda$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding $ lambda$ extension of the two-point diagonal correlation function, $ C(N,N; lambda)$, is shown, for arbitrary $lambda$, to be a solution of the sigma form of the Painlev{e} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $ f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $ E$. Each $ f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $ E$ and $ K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll structure. The previous $ lambda$-extensions, $ C(N,N; lambda)$ are, for singled-out values $ lambda= cos(pi m/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painleve VI are actually algebraic functions, being associated with modular curves.
The lace expansion for the Ising two-point function was successfully derived in Sakai (Commun. Math. Phys., 272 (2007): 283--344). It is an identity that involves an alternating series of the lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster $x$-space decay (as the two-point function cubed) above the critical dimension $d_c$ ($=4$ for finite-variance models), if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of Lemma 4.2 in Sakai (2007), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic Lemma 4.2 of Sakai (2007), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in Proposition 4.1 of Sakai (2007) but nonetheless obey the same fast decay above the critical dimension $d_c$. Consequently, the lace-expansion results for the Ising and $varphi^4$ models so far are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients.
We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions $chi_{d}^{(1)}$ and $chi_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ${chi}^{(5)}_d(t)$, but only modulo a given prime. We use these exact linear differential equations to show that, not only the russian-doll structure, but also the direct sum structure on the linear differential operators for the $ n$-particle contributions $chi_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $ f^{(n)}$. We show that the $ n^{th}$ particle contributions $chi_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|sinh2E_v/kT sinh 2E_h/kT|=1$ as $ nto infty$.
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