An elementary approach to the Szegő theorems

Abstract

The strong Szegő theorem and the Fisher-Hartwig formula describe the asymptotic behavior of determinants obtained by restricting translationally invariant operators to large finite regions. In particular, Szegő's theorem states that, beyond the leading extensive contribution to such a determinant, there is a universal finite correction, $\sum_{r>0} rV_rV_{-r}$ where $V_r$ are the Fourier coefficients of the logarithm of the symbol. Although this formula appears throughout analysis and mathematical physics, the origin of its quadratic form and of the weight $r$ is often obscured in its standard analytic derivations. We develop an elementary picture in which the strong Szegő term arises from boundary leakage: paths contributing to the matrix exponential leave the finite interval and subsequently return. Counting the shortest such excursions immediately produces the factor $r$. We then show that an isolated boundary contributes exactly one half of the strong Szegő correction, while the two boundaries of a long interval decouple. For finite-range matrices this gives a rigorous and intuitive elementary derivation of the asymptotics, with extensions to sufficiently rapidly decaying interactions. A complementary free-fermion argument provides a physical interpretation of the same cancellation mechanism. Numerical examples illustrate the asymtotics in the well controlled regime, the Fisher Hartwig logarithmic correction. The boundary formulation does not itself require translation invariance and therefore suggests natural extensions to disordered systems, higher dimensions, and symbols with internal matrix structure. To illustrate this we numerically go beyond the formal applicability region of the Szegő formula, considering slow decay and disordered cases, and showing in some cases a robustness of the approach even for slowly decaying but disordered potentials.