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

DOI10.1214/21-AIHP1211zbMATH Open1494.30074arXiv2101.03557OpenAlexW3199713517MaRDI QIDQ2157444

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Publication date: 22 July 2022

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Abstract: We rigorously compute the integrable system for the limiting $(Nightarrowinfty)$ distribution function of the extreme momentum of $N$ noninteracting fermions when confined to an anharmonic trap $V(q)=q^{2n}$ for $ninmathbb{Z}_{geq1}$ 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. cite{LW,ACQ}. For general $ngeq2$, a novel higher order finite temperature Airy kernel has recently emerged in physics literature cite{DMS} 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.

Full work available at URL: https://arxiv.org/abs/2101.03557

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