2015/08/12 by Moll, Alexander · 2 citations
#Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1508.03063
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 dF^ηNS( 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 dFλ(c | ε2, ε1) the Rayleigh measure of the profile of a partition λ of anisotropy (ε2, ε1) ∈ ℂ2 for ℏ = - ε1 ε2, ε= ε1 + ε2 invariant under ε2 \longleftrightarrow ε1. 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 Υv ( ⋅ | 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) - \tfrac12 dim \mathbbT = + \tfrac12 - \tfrac12 = 0.