Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel

We rigorously compute the integrable system for the limiting (N → ∞) distribution function of the extreme momentum of N noninteracting fermions when confined to an anharmonic trap V (q) = q2n for n ∈ Z≥1 at positive temperature. More precisely, the edge momentum statistics in the harmonic trap n = 1 are known to obey the weak asymmetric KPZ crossover law which is realized via the finite temperature Airy kernel determinant or equivalently via a Painlev ́e-II integro-differential transcendent, cf. [3,35]. For general n ≥ 2, a novel higher order finite temperature Airy kernel has recently emerged in physics literature [33] and we show that the corresponding edge law in momentum space is now governed by a distinguished Painlev ́e-II integro-differential hierarchy. Our analysis is based on operator-valued Riemann-Hilbert techniques which produce a Lax pair for an operator-valued Painlev ́e-II ODE system that naturally encodes the aforementioned hierarchy. As byproduct, we establish a connection of the integro-differential Painlev ́e-II hierarchy to a novel integro-differential mKdV hierarchy.