2021/01/10 by Bothner, Thomas, Cafasso, Mattia, Tarricone, Sofia · 1 citation
#35J10 #42A38 #81V70 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 45J05 #Probability (math.PR) #Quantum Gases (cond-mat.quant-gas) #Secondary 30E25 #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2101.03557
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 Painlevé-II integro-differential transcendent, cf. \citeLW,ACQ. For general n≥ 2, a novel higher order finite temperature Airy kernel has recently emerged in physics literature \citeDMS 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.