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arXiv · 2011.14561

Asymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations

Estelle Basor·Torsten Ehrhardt·Roozbeh Gharakhloo·Alexander Its·Yuqi Li
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Abstract

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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Estelle Basor, Torsten Ehrhardt, Roozbeh Gharakhloo, Alexander Its, Yuqi Li. 2020-11-30. Asymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations. https://arxiv.org/abs/2011.14561

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