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Register a new userAsymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations
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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.
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In this article, we study the large $n$ asymptotic expansions of $ntimes n$ Toeplitz determinants whose symbols are indicator functions of unions of arc-intervals of the unit circle. In particular, we use an Hermitian matrix model reformulation of the problem to provide a rigorous derivation of the general form of the large $n$ expansion when the symbol is an indicator function of either a single arc-interval or several arc-intervals with a discrete rotational symmetry. Moreover, we prove that the coefficients in the expansions can be reconstructed, up to some constants, from the Eynard-Orantin topological recursion applied to some explicit spectral curves. In addition, when the symbol is an indicator function of a single arc-interval, we provide the corresponding normalizing constants using a Selberg integral and illustrate the theoretical results with numeric simulations up to order $oleft(frac{1}{n^4}right)$. We also briefly discuss the situation when the number of arc-intervals increases with $n$, as well as more general Toeplitz determinants to which we may apply the present strategy.
We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painleve V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results.
Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017)], we show that the partition function of the anisotropic square lattice Ising model on the $L times M$ rectangle, with open boundary conditions in both directions, is given by the determinant of a $M/2 times M/2$ Hankel matrix, that equivalently can be written as the Pfaffian of a skew-symmetric $M times M$ Toeplitz matrix. The $M-1$ independent matrix elements of both matrices are Fourier coefficients of a certain symbol function, which is given by the ratio of two characteristic polynomials. These polynomials are associated to the different directions of the system, encode the respective boundary conditions, and are directly related through the symmetry of the considered Ising model under exchange of the two directions. The results can be generalized to other boundary conditions and are well suited for the analysis of finite-size scaling functions in the critical scaling limit using SzegH{o}s theorem.
We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
We give the exact expressions of the partial susceptibilities $chi^{(3)}_d$ and $chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],, [1,1];, z)$ and $_4F_3([1/2,1/2,1/2,1/2],, [1,1,1]; , z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $chi^{(3)}_d$ and $chi^{(4)}_d$. We also give new results for $chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only Derived from Geometry (globally nilpotent), but actually correspond to Special Geometry (homomorphic to their formal adjoint). This raises the question of seeing if these special geometry Ising-operators, are special ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.
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