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

TL;DR

This paper establishes the asymptotic behavior of bordered Toeplitz determinants and applies these results to rigorously analyze next-to-diagonal correlations in the anisotropic square lattice Ising model, confirming their long-range order in the low temperature regime.

Contribution

It proves a strong Szegő limit theorem analogue for bordered Toeplitz determinants and applies it to Ising model correlations, providing rigorous justification and asymptotic analysis.

Findings

Next-to-diagonal correlations exhibit the same long-range order as diagonal ones at low temperature.

The anisotropy affects the subleading term in the asymptotics of correlations.

The results are obtained using Riemann-Hilbert and operator theory techniques.

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.