The Multiplicative Constant in Asymptotics of Higher-Order Analogues of the Tracy–Widom Distribution

In this paper, we are concerned with higher-order analogs of the Tracy–Widom distribution, which describe the eigenvalue distributions in unitary random matrix models near critical edge points. The associated kernels are constructed by functions related to the even members of the Painlevé I hierarchy [Formula: see text], and are regarded as higher-order analogs of the Airy kernel. By performing a uniform asymptotic analysis on the associated Riemann–Hilbert problem, we derive the multiplicative constant in the large gap asymptotics of the distribution, resolving an open problem in the work of Claeys, Its and Krasovsky for special choice of parameters. An important new feature of the expression is the involvement of an integral of the Hamiltonian associated with a special, real, pole-free solution for [Formula: see text]. In addition, we show that the total integral of the Hamiltonian vanishes for all [Formula: see text] and establish a transition from the higher-order Tracy–Widom distribution to the classical one in the asymptotic regime. Our approach also works for general parameters and can be adapted to calculate similar critical constants in other problems arising from mathematical physics.