Asymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations
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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2020
- DOI:
- 10.48550/arXiv.2011.14561
- arXiv:
- arXiv:2011.14561
- Bibcode:
- 2020arXiv201114561B
- Keywords:
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- Mathematical Physics
- E-Print:
- 44 pages