Counting free fermions on a line: A Fisher-Hartwig asymptotic expansion for the Toeplitz determinant in the double-scaling limit

We derive an asymptotic expansion for a Wiener-Hopf determinant arising in the problem of counting one-dimensional free fermions on a line segment at zero temperature. This expansion is an extension of the result in the theory of Toeplitz and Wiener-Hopf determinants known as the generalized Fisher-Hartwig conjecture. The coefficients of this expansion are conjectured to obey certain periodicity relations, which renders the expansion explicitly periodic in the 'counting parameter'. We present two methods to calculate these coefficients and verify the periodicity relations order by order: the matrix Riemann-Hilbert problem and the Painlevé V equation. We show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients.