Random Partitions and the Quantum Benjamin-Ono Hierarchy [article]

Alexander Moll
2017 arXiv   pre-print
We derive exact and asymptotic results for random partitions from general results in the semi-classical analysis of coherent states applied to the classical periodic Benjamin-Ono equation at critical regularity s= -1/2. We find classical dF_ |v (c| ε) and quantum dF̂^η_NS( c | ħ, ε)|_Ψ conserved densities for this system with dispersion coefficient ε extending Nazarov-Sklyanin (2013). For quantum stationary states, this conserved density is dF_λ(c | ε_2, ε_1) the Rayleigh measure of the profile
more » ... of a partition λ of anisotropy (ε_2, ε_1) ∈C^2 for ħ = - ε_1 ε_2, ε= ε_1 + ε_2 invariant under ε_2 ε_1. As Jack polynomials are the quantum stationary states and Stanley's Cauchy kernel (1989) is the reproducing kernel, the random values of the quantum periodic Benjamin-Ono hierarchy in a coherent state Υ_v ( · | ħ) are a "Jack measure" on partitions, a dispersive generalization of Okounkov's Schur measures (1999). By our general results for coherent states, we have concentration on a limit shape as ħ→ 0, the classical conserved density at v, and quantum fluctuations are an explicit Gaussian field. Our results follow from an enumerative asymptotic expansion in ħ and ε of joint cumulants over new combinatorial objects we call "ribbon paths". Our results reflect the fact that at fixed ħ>0 the weight defining Fock space is already a fractional Brownian motion of variance ħ and Hurst index (-s) - 12T = + 12 - 12 = 0.
arXiv:1508.03063v3 fatcat:agw5xrfmfrcj3kkfjxtn5zzinm