Asymptotics of Bordered Toeplitz Determinants and Next-to-Diagonal Ising Correlations

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We prove the analogue of the strong Szegő limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk (Physica A 144:44–104, 1987) for the next-to-diagonal correlations \(\langle \sigma _{0,0}\sigma _{N-1,N} \rangle \) in the 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. We also confirm the leading and subleading terms in an asymptotic formula of Cheng and Wu (Phys Rev 164:719–735, 1967) for \(\langle \sigma _{0,0}\sigma _{M,N} \rangle \) when \(M=N\) and \(M=N-1\), thereby establishing the anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.