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Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel. (English) Zbl 1494.30074

Summary: We rigorously compute the integrable system for the limiting \((N\to \infty )\) distribution function of the extreme momentum of \(N\) noninteracting fermions when confined to an anharmonic trap \(V(q)={q^{2n}}\) for \(n\in{\mathbb{Z}_{\ge 1}}\) 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 Painlevé-II integro-differential transcendent, cf. [K. Liechty and D. Wong, Ann. Inst. Henri Poincaré, Probab. Stat. 56, No. 2, 1072–1098 (2020; Zbl 1447.82014); G. Amir et al., Commun. Pure Appl. Math. 64, No. 4, 466–537 (2011; Zbl 1222.82070)]. For general \(n\ge 2\), a novel higher order finite temperature Airy kernel has recently emerged in physics literature [P. Le Doussal, S. N. Majumdar and G. Schehr, “Multicritical edge statistics for the momenta of fermions in nonharmonic traps”, Phys. Rev. Lett. 121 (3) (2018) 030603, https://doi.org/10.1103/PhysRevLett.121.030603] and we show that the corresponding edge law in momentum space is now governed by a distinguished Painlevé-II integro-differential hierarchy. Our analysis is based on operator-valued Riemann-Hilbert techniques which produce a Lax pair for an operator-valued Painlevé-II ODE system that naturally encodes the aforementioned hierarchy. As byproduct, we establish a connection of the integro-differential Painlevé-II hierarchy to a novel integro-differential mKdV hierarchy.

MSC:

30E25 Boundary value problems in the complex plane
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
45J05 Integro-ordinary differential equations
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