Asymptotics of the Airy-kernel determinant
Let \(K_s\) be the trace-class operator with kernel \[ K_s(t,u)=\frac{\text{Ai}(t)\text{Ai}'(u)-\text{Ai}(u)\text{Ai}'(t)}{t-u} \] acting on \(L^2(-s,\infty)\), where \(\text{Ai}(x)\) is the Airy function. The paper is concerned with the behavior of the Fredholm determinant \(\text{det}(I-K_s)\) as \(s\rightarrow +\infty\). In fact, the authors prove that the large \(s\)-asymptotic behavior of this determinant is given by \[ \ln \text{det}(I-K_s)=-\frac{s^3}{12}-\frac{1}{8}\ln s+\chi +O(s^{-3/2}), \] where \[ \chi=\frac{1}{24}\ln 2+\zeta'(-1) \] and \(\zeta(s)\) is the Riemann zeta-function. The value of the constant \(\chi\) is computed using a Riemann-Hilbert approach.
scientific article