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• W2212778857 abstract "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 $dF_{star |v} (c| overline{varepsilon})$ and quantum $dhat{F}^{eta_{NS}}( c | hbar, overline{varepsilon})|_{Psi}$ conserved densities for this system with dispersion coefficient $overline{varepsilon}$ extending Nazarov-Sklyanin (2013). For quantum stationary states, this conserved density is $dF_{lambda}(c | varepsilon_2, varepsilon_1)$ the Rayleigh of the profile of a partition $lambda$ of anisotropy $(varepsilon_2, varepsilon_1) in mathbb{C}^2$ for $hbar = - varepsilon_1 varepsilon_2$, $overline{varepsilon}= varepsilon_1 + varepsilon_2$ invariant under $varepsilon_2 longleftrightarrow varepsilon_1$. As 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 $Upsilon_v ( cdot | hbar)$ 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 $hbar rightarrow 0$, the classical conserved density at $v$, and quantum fluctuations are an explicit Gaussian field. Our results follow from an enumerative asymptotic expansion in $hbar$ and $overline{varepsilon}$ of joint cumulants over new combinatorial objects we call ribbon paths. Our results reflect the fact that at fixed $hbar>0$ the weight defining Fock space is already a fractional Brownian motion of variance $hbar$ and Hurst index $(-s) - tfrac{1}{2} dim mathbb{T} = + tfrac{1}{2} - tfrac{1}{2} = 0.$" @default.
• W2212778857 created "2016-06-24" @default.
• W2212778857 creator A5059633067 @default.
• W2212778857 date "2016-01-01" @default.
• W2212778857 modified "2023-09-27" @default.
• W2212778857 title "Random partitions and the quantum Benjamin-Ono hierarchy" @default.
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• W2212778857 hasPublicationYear "2016" @default.
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